a synchronous and iterative flux correction formalism for coupled transport equations many problems of fluid dynamics involve the coupled transport of several density like dependent variables for instance densities of mass and momenta in elastic flows in this paper a conservative and synchronous flux corrected transport fct formalism is developed which aims at a consistent transport of such variables the technique differs from traditional fct algorithms in two respects first the limiting of transportive fluxes of the primary variables e g mass and momentum does not derive from smooth estimates of the variables but it derives from analytic constraints implied by the lagrangian form of the governing continuity equations which are imposed on the specific mixing ratios of the variables e g velocity components second the traditional fct limiting based on sufficiency conditions is augmented by an iterative procedure which approaches the necessity requirements this procedure can also be used in the framework of traditional fct schemes and a demonstration is provided that it can significantly reduce some of the pathological behaviors of fct algorithms although the approach derived is applicable to the transport of arbitrary conserved quantities it is particularly useful for the synchronous transport of mass and momenta in elastic flows where it assures intrinsic stability of the algorithm regardless of the magnitude of the mass density variable this latter property becomes especially important in fluids with large density variations or in models with a material vertical coordinate e g geophysical hydrostatic stratified flows in isopycnic isentropic coordinates where material surfaces can collapse to zero mass layers admitting therefore arbitrarily large local courant numbers

a staggered semi implicit discontinuous galerkin method for the two dimensional incompressible navier stokes equations in this paper we propose a new spatially high order accurate semi implicit discontinuous galerkin dg method for the solution of the two dimensional incompressible navier stokes equations on staggered unstructured curved meshes while the discrete pressure is defined on the primal grid the discrete velocity vector field is defined on an edge based dual grid the flexibility of high order dg methods on curved unstructured meshes allows to discretize even complex physical domains on rather coarse grids formal substitution of the discrete momentum equation into the discrete continuity equation yields one sparse linear equation system with four non zero blocks per element for only one scalar unknown namely the pressure the method is computationally efficient since the resulting system is not only very sparse but also symmetric and positive definite for appropriate boundary conditions furthermore all the volume and surface integrals needed by the scheme presented in this paper depend only on the geometry and the polynomial degree of the basis and test functions and can therefore be precomputed and stored in a preprocessor stage which leads to savings in terms of computational effort for the time evolution part in this way also the extension to a fully curved isoparametric approach becomes natural and affects only the preprocessing step the method is validated for polynomial degrees up to p 3 by solving some typical numerical test problems and comparing the numerical results with available analytical solutions or other numerical and experimental reference data

a note on the design of hp adaptive finite element methods for elliptic partial differential equations we introduce an hp adaptive finite element algorithm based on a combination of reliable and efficient residual error indicators and a new hp extension control technique which assesses the local regularity of the underlying analytical solution on the basis of its local legendre series expansion numerical experiments confirm the robustness and reliability of the proposed algorithm

a second order characteristics finite element scheme for natural convection problems in this paper a second order characteristics finite element scheme is applied to the numerical solution of natural convection problems firstly after recalling the mathematical model a second order time discretization of the material time derivative is introduced next fully discretized schemes are proposed by using finite element methods numerical results for the two dimensional problem of buoyancy driven flow in a square cavity with differentially heated side walls are given and compared with a reference solution

asymptotic preserving imex finite volume schemes for low mach number euler equations with gravitation in this paper we will present and analyze a new class of the imex finite volume schemes for the euler equations with a gravity source term we will in particular concentrate on a singular limit of weakly compressible flows when the mach number m1 in order to efficiently resolve slow dynamics we split the whole nonlinear system in a stiff linear part governing the acoustic and gravity waves and a non stiff nonlinear part that models nonlinear advection effects for time discretization we use a special class of the so called globally stiffly accurate imex schemes and approximate the stiff linear operator implicitly and the non stiff nonlinear operator explicitly for spatial discretization the finite volume approximation is used with the central and rusanov laxfriedrichs numerical fluxes for the linear and nonlinear subsystem respectively in the case of a constant background potential temperature we prove theoretically that the method is asymptotically consistent and asymptotically stable uniformly with respect to small mach number we also analyze experimentally convergence rates in the singular limit when the mach number tends to zero

optimizing the geometrical accuracy of curvilinear meshes this paper presents a method to generate valid high order meshes with optimized geometrical accuracy the high order meshing procedure starts with a linear mesh that is subsequently curved without taking care of the validity of the high order elements an optimization procedure is then used to both untangle invalid elements and optimize the geometrical accuracy of the mesh standard measures of the distance between curves are considered to evaluate the geometrical accuracy in planar two dimensional meshes but they prove computationally too costly for optimization purposes a fast estimate of the geometrical accuracy based on taylor expansions of the curves is introduced an unconstrained optimization procedure based on this estimate is shown to yield significant improvements in the geometrical accuracy of high order meshes as measured by the standard hausdorff distance between the geometrical model and the mesh several examples illustrate the beneficial impact of this method on cfd solutions with a particular role of the enhanced mesh boundary smoothness

analysis of adaptive mesh refinement for imex discontinuous galerkin solutions of the compressible euler equations with application to atmospheric simulations the resolutions of interests in atmospheric simulations require prohibitively large computational resources adaptive mesh refinement amr tries to mitigate this problem by putting high resolution in crucial areas of the domain we investigate the performance of a tree based amr algorithm for the high order discontinuous galerkin method on quadrilateral grids with non conforming elements we perform a detailed analysis of the cost of amr by comparing this to uniform reference simulations of two standard atmospheric test cases density current and rising thermal bubble the analysis shows up to 15 times speed up of the amr simulations with the cost of mesh adaptation below 1 of the total runtime we pay particular attention to the implicit explicit imex time integration methods and show that the ark2 method is more robust with respect to dynamically adapting meshes than bdf2 preliminary analysis of preconditioning reveals that it can be an important factor in the amr overhead the compiler optimizations provide significant runtime reduction and positively affect the effectiveness of amr allowing for speed ups greater than it would follow from the simple performance model

semi implicit discontinuous galerkin methods for the incompressible navier stokes equations on adaptive staggered cartesian grids in this paper a new high order semi implicit discontinuous galerkin method si dg is presented for the solution of the incompressible navier stokes equations on staggered space time adaptive cartesian grids amr in two and three space dimensions the pressure is written in the form of piecewise polynomials on the main grid which is dynamically adapted within a cell by cell amr framework according to the time dependent main grid different face based spatially staggered dual grids are defined for the piece wise polynomials of the respective velocity components although the resulting adaptive staggered grids are more complex than classical uniform cartesian meshes the numerical scheme can still be written in a rather compact form thanks to the use of a tensor product formulation for the definition of the nodal basis in the d dimensional space d 2 3 all the discrete operators can be efficiently written as a combination of linear one dimensional operators acting in the d space directions separately arbitrary high order of accuracy is achieved in space while a very simple semi implicit time discretization is obtained via an explicit discretization of the nonlinear convective terms and an implicit discretization of the pressure gradient in the momentum equation and of the divergence of the velocity field in the continuity equation the real advantages of the staggered grid arise in the solution of the schur complement associated with the saddle point problem of the discretized incompressible navier stokes equations i e after substituting the discrete momentum equations into the discrete continuity equation this leads to a linear system for only one unknown the scalar pressure indeed the resulting linear pressure system is shown to be symmetric and positive definite in order to avoid a quadratic stability condition for the parabolic terms given by the viscous stress tensor an implicit discretization is also used for the diffusive terms in the momentum equation a particular feature of our staggered dg approach is that the viscous stress tensor is discretized on the dual mesh this corresponds to the use of a lifting operator but on the staggered grid both linear systems for pressure and velocity are very efficiently solved by means of a classical matrix free conjugate gradient method for which fast convergence is observed moreover it should be noticed that all test cases shown in this paper have been performed without the use of any preconditioner due to the explicit discretization of the nonlinear convective terms the final algorithm is stable within the classical cfl type time step restriction for explicit dg methods based on the fluid velocity moreover by using a suitable local time stepping technique based on the ader dg framework each element can use its optimal local time step also within the amr context the new space time adaptive staggered dg scheme has been thoroughly verified for polynomial degrees up to n 9 for a large set of non trivial test problems in two and three space dimensions for which analytical numerical or experimental reference solutions exist the high order of accuracy of the scheme is shown by means of a numerical convergence table obtained by comparing the numerical solution against a smooth analytical solution to the knowledge of the authors this is the first staggered semi implicit dg scheme for the incompressible navier stokes equations on space time adaptive meshes in two and three space dimensions

a discontinuous galerkin method for the shallow water equations in spherical triangular coordinates a global model of the atmosphere is presented governed by the shallow water equations and discretized by a runge kutta discontinuous galerkin method on an unstructured triangular grid the shallow water equations on the sphere a two dimensional surface in r 3 are locally represented in terms of spherical triangular coordinates the appropriate local coordinate mappings on triangles on every triangular grid element this leads to a two dimensional representation of tangential momentum and therefore only two discrete momentum equations the discontinuous galerkin method consists of an integral formulation which requires both area elements and line element faces integrals here we use a rusanov numerical flux to resolve the discontinuous fluxes at the element faces a strong stability preserving third order runge kutta method is applied for the time discretization the polynomial space of order k on each curved triangle of the grid is characterized by a lagrange basis and requires high order quadature rules for the integration over elements and element faces for the presented method no mass matrix inversion is necessary except in a preprocessing step the validation of the atmospheric model has been done considering standard tests from williamson et al d l williamson j b drake j j hack r jakob p n swarztrauber a standard test set for numerical approximations to the shallow water equations in spherical geometry j comput phys 102 1992 211 224 unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream a convergence rate of o dx k 1 was observed in the model experiments furthermore a numerical experiment is presented for which the third order time integration method limits the model error thus the time step dt is restricted by both the cfl condition and accuracy demands conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k

variational multiscale stabilization of high order spectral elements for the advection diffusion equation one major issue in the accurate solution of advection dominated problems by means of high order methods is the ability of the solver to maintain monotonicity this problem is critical for spectral elements where gibbs oscillations may pollute the solution however typical filter based stabilization techniques used with spectral elements are not monotone in this paper residual based stabilization methods originally derived for finite elements are constructed and applied to high order spectral elements in particular we show that the use of the variational multiscale vms method greatly improves the solution of the transport diffusion equation by reducing over and under shoots and can be therefore considered an alternative to filter based schemes we also combine these methods with discontinuity capturing schemes dc to suppress oscillations that may occur in proximity of boundaries or internal layers additional improvement in the solution is also obtained when a method that we call fos for first order subcells is used in combination with vms and dc in the regions where discontinuities occur fos subdivides a spectral element of order p into p 2 subcells and then uses 1st order basis functions and integration rules on every subcell of the element the algorithms are assessed with the solution of classical steady and transient 1d 2d and pseudo 3d problems using spectral elements up to order 16

a semi implicit semi lagrangian p adaptive discontinuous galerkin method for the shallow water equations a semi implicit and semi lagrangian discontinuous galerkin method for the shallow water equations is proposed for applications to geophysical scale flows a non conservative formulation of the advection equation is employed in order to achieve a more treatable form of the linear system to be solved at each time step the method is equipped with a simple p adaptivity criterion that allows to adjust dynamically the number of local degrees of freedom employed to the local structure of the solution numerical results show that the method captures well the main features of gravity and inertial gravity waves as well as reproducing correct solutions in nonlinear test cases with analytic solutions the accuracy and effectiveness of the method are also demonstrated by numerical results obtained at high courant numbers and with automatic choice of the local approximation degree

semi implicit finite difference methods for the two dimensional shallow water equation abstract in this paper a semi implicit finite difference method for the 2 dimensional shallow water equations is derived and discussed a characteristic analysis of the governing equations is carried out first in order to determine those terms to be discretized implicitly so that the stability of the method will not depend upon the celerity such terms are the gradient of the water surface elevation in the momentum equations and the velocity divergence in the continuity equation the convective terms are discretized explicitly the simpler explicit discretization for the convective terms is the upwind discretization which is conditionally stable and introduces some artificial viscosity it is shown that the stability restriction is eliminated and the artificial viscosity is reduced when an eulerian lagrangian approach with large time steps is used to discretize the convective terms this method at each time step requires the solution of a linear symmetric 5 diagonal system such a system is diagonally dominant with positive elements on the main diagonal and negative ones elsewhere thus existence and uniqueness of the numerical solution is assured the resulting algorithm is mass conservative and fully vectorizable for an efficient implementation on modern vector computers the performance of this method is further improved when used in combination with an adi technique which results in two sets of simpler linear 3 diagonal systems and maintains all the properties described above

adjoint based airfoil optimization with adaptive isogeometric discontinuous galerkin method abstract in this work an adjoint based airfoil shape optimization algorithm is developed based on the adaptive isogeometric discontinuous galerkin method for compressible euler equations to investigate the significance of each design variable of airfoil b spline parameterization we first parameterize the airfoil by b spline curve approximation with some control points viewed as design variables and build the b spline representation of the flow field with the curve to apply the goal oriented h adaptive isogeometric dg method for flow solution then we compute and employ the discrete adjoint solutions for both multi target error estimation in adaptive mesh refinement with the isogeometric nature not only all the geometrical cells but also the numerical basis functions can be analytically expressed by the design variables indicating that the numerical solutions and objective could be differentiable with respect to those variables consequently the gradient is totally computed in an accurate approach and the sensitivity analysis is thus improved by reducing the spatial discretization error and introducing the analytical expression of derivative to reveal the key parameters for optimization in an intuitive and efficient manner although the sqp optimization algorithm is adopted in the paper the given accurate gradient can be applied to any gradient based optimization algorithm the proposed algorithm is demonstrated on rae2822 airfoil with inviscid transonic flow where the shape is optimized to minimize the drag coefficient at a constrained lift and airfoil area the numerical results show that the drag is much more sensitive to the design variables near the tailing edge at the beginning but sensitivity is reduced when optimal

a comparison of refinement indicators for p adaptive simulations of steady and unsteady flows using discontinuous galerkin methods abstract this paper presents an analysis of refinement indicators for the simulation of steady and unsteady flows by means of p adaptive algorithms for discontinuous galerkin dg methods residual error discretization error and feature based indicators are compared by studying their effect on convergence history computational gain and refinement regions selected by the adaptive algorithm the analysis is initially carried out on steady flow configurations static p adaptive simulations of the periodic flow past a cylinder at r e 100 are then performed compared to uniform p refinement a reduction between 50 and 75 in the number of degrees of freedom is obtained for all test cases considered the accuracy and efficiency observed for the small scale energy density 1 and spectral decay 2 indicators demonstrate their great potential for the adaptive simulation of unsteady flows

comparison between adaptive and uniform discontinuous galerkin simulations in dry 2d bubble experiments adaptive mesh refinement generally aims to increase computational efficiency without compromising the accuracy of the numerical solution however it is an open question in which regions the spatial resolution can actually be coarsened without affecting the accuracy of the result this question is investigated for a specific example of dry atmospheric convection namely the simulation of warm air bubbles for this purpose a novel numerical model is developed that is tailored towards this specific application the compressible euler equations are solved with a discontinuous galerkin method time integration is done with an imex method and the dynamic grid adaptivity uses space filling curves via the amatos function library so far the model is able to simulate dry flow in two dimensional geometry without subgrid scale modeling the model is tested with three standard test cases an error indicator is introduced for a warm air bubble test case which allows one to compare the accuracy between different choices of refinement regions without knowing the exact solution essentially this is done by comparing features of the solution that are strongly sensitive to spatial resolution for the rising warm air bubble the additional error by using adaptivity is smaller than 1 of the total numerical error if the average number of elements used for the adaptive simulation is about a factor of two times smaller than the number used for the simulation with the uniform fine resolution grid correspondingly the adaptive simulation is almost two times faster than the uniform simulation furthermore the adaptive simulation is more accurate than a uniform simulation when both use the same cpu time

continuous and discontinuous galerkin methods for a scalable three dimensional nonhydrostatic atmospheric model limited area mode this paper describes a unified element based galerkin ebg framework for a three dimensional nonhydrostatic model for the atmosphere in general ebg methods possess high order accuracy geometric flexibility excellent dispersion properties and good scalability our nonhydrostatic model based on the compressible euler equations is appropriate for both limited area and global atmospheric simulations both a continuous galerkin cg or spectral element and discontinuous galerkin dg model are considered using hexahedral elements the formulation is suitable for both global and limited area atmospheric modeling although we restrict our attention to 3d limited area phenomena in this study global atmospheric simulations will be presented in a follow up paper domain decomposition and communication algorithms used by both our cg and dg models are presented the communication volume and exchange algorithms for cg and dg are compared and contrasted numerical verification of the model was performed using two test cases flow past a 3d mountain and buoyant convection of a bubble in a neutral atmosphere these tests indicate that both cg and dg can simulate the necessary physics of dry atmospheric dynamics scalability of both methods is shown up to 8192 cpu cores with near ideal scaling for dg up to 32 768 cores

an isoparametric approach to high order curvilinear boundary layer meshing abstract the generation of high order curvilinear meshes for complex three dimensional geometries is presently a challenging topic particularly for meshes used in simulations at high reynolds numbers where a thin boundary layer exists near walls and elements are highly stretched in the direction normal to flow in this paper we present a conceptually simple but very effective and modular method to address this issue we propose an isoparametric approach whereby a mesh containing a valid coarse discretization comprising of high order triangular prisms near walls is refined to obtain a finer prismatic or tetrahedral boundary layer mesh the validity of the prismatic mesh provides a suitable mapping that allows one to obtain very fine mesh resolutions across the thickness of the boundary layer we describe the method in detail for a high order approximation using modal basis functions discuss the requirements for the splitting method to produce valid prismatic and tetrahedral meshes and provide a sufficient criterion of validity in both cases by considering two complex aeronautical configurations we demonstrate how highly stretched meshes with sufficient resolution within the laminar sublayer can be generated to enable the simulation of flows with reynolds numbers of 10 6 and above

exact projections and the lagrange galerkin method a realistic alternative to quadrature abstract the lagrange galerkin method has proved a very successful method in computational fluid dynamics cfd in practice but it has suffered from question marks about its stability when the integrals are approximated by quadratures in this paper we will prove instability for wider ranges of quadrature formulae than previously considered we will also introduce on approximate integration technique on triangular grids that cheaply recovers the unconditional stability of the lagrange galerkin method with no loss in accuracy the same technique can also be used for the exact projection of data from one arbitrary triangular grid to another different arbitrary triangular grid this could be used in grid adaptation algorithms or in triangular multigrid procedures

data assimilation for linear parabolic equations minimax projection method in this paper we propose a state estimation method for linear parabolic partial differential equations pde that accounts for errors in the model truncation and observations it is based on an extension of the galerkin projection method the extended method models projection coefficients representing the state of the pde in some basis by means of a differential algebraic equation dae the original estimation problem for the pde is then recast as a state estimation problem for the constructed dae using a linear continuous minimax filter we construct a numerical time integrator that preserves the monotonic decay of a nonstationary lyapunov function along the solution to conclude we demonstrate the efficacy of the proposed method by applying it to the tracking of a discharged pollutant slick in a two dimensional fluid

a staggered space time discontinuous galerkin method for the three dimensional incompressible navier stokes equations on unstructured tetrahedral meshes in this paper we propose a novel arbitrary high order accurate semi implicit space time discontinuous galerkin method for the solution of the three dimensional incompressible navier stokes equations on staggered unstructured curved tetrahedral meshes as is typical for space time dg schemes the discrete solution is represented in terms of space time basis functions this allows to achieve very high order of accuracy also in time which is not easy to obtain for the incompressible navier stokes equations similarly to staggered finite difference schemes in our approach the discrete pressure is defined on the primary tetrahedral grid while the discrete velocity is defined on a face based staggered dual grid while staggered meshes are state of the art in classical finite difference schemes for the incompressible navier stokes equations their use in high order dg schemes is still quite rare a very simple and efficient picard iteration is used in order to derive a space time pressure correction algorithm that achieves also high order of accuracy in time and that avoids the direct solution of global nonlinear systems formal substitution of the discrete momentum equation on the dual grid into the discrete continuity equation on the primary grid yields a very sparse five point block system for the scalar pressure which is conveniently solved with a matrix free gmres algorithm from numerical experiments we find that the linear system seems to be reasonably well conditioned since all simulations shown in this paper could be run without the use of any preconditioner even up to very high polynomial degrees for a piecewise constant polynomial approximation in time and if pressure boundary conditions are specified at least in one point the resulting system is in addition symmetric and positive definite this allows us to use even faster iterative solvers like the conjugate gradient method the flexibility and accuracy of high order space time dg methods on curved unstructured meshes allows to discretize even complex physical domains with very coarse grids in both space and time the proposed method is verified for approximation polynomials of degree up to four in space and time by solving a series of typical 3d test problems and by comparing the obtained numerical results with available exact analytical solutions or with other numerical or experimental reference data to the knowledge of the authors this is the first time that a space time discontinuous galerkin finite element method is presented for the three dimensional incompressible navier stokes equations on staggered unstructured tetrahedral grids high order staggered space time dg scheme for the 3d incompressible navier stokes equations primal grid based on curved tetrahedra dual grid based on non standard five point hexahedra new discretization of the viscous terms on the main grid using lifting operators on the dual grid very simple space time pressure correction algorithm providing also high order in time all linear systems have maximum sparsity and use the smallest possible optimal five point stencil

a free lagrange finite element method using space time elements we consider the construction and implementation of a free lagrange finite element method where local mesh modifications can be introduced during computation the method is based on the use of sp

numerical analysis of a second order pure lagrange galerkin method for convection diffusion problems part ii fully discretized scheme and numerical results we analyze a second order pure lagrange galerkin method for variable coefficient convection diffusion possibly degenerate diffusion equations with mixed dirichlet robin boundary conditions in a previous paper the proposed second order pure lagrangian time discretization scheme was introduced and analyzed for the same problem more precisely the l infty h 1 stability and l infty h 1 error estimates of order o delta t 2 had been obtained moreover for the particular case of incompressible flows stability inequalities with constants independent of the final time have been stated in the present paper l infty h 1 error estimates of order o delta t 2 o h k are obtained for the fully discretized pure lagrange galerkin method to prove these results we use some properties obtained in the previous paper finally numerical tests are presented that confirm the theoretical results

stability accuracy and efficiency of a semi implicit method for three dimensional shallow water flow abstract the stability analysis the accuracy and the efficiency of a semi implicit finite difference scheme for the numerical solution of a three dimensional shallow water model are presented and discussed the governing equations are the three dimensional reynolds equations in which pressure is assumed to be hydrostatic the pressure gradient in the momentum equations and the velocities in the vertically integrated continuity equation are discretized with the method with being an implicitness parameter it is shown that the method is stable for 1 2 1 unstable for 1 2 and highest accuracy and efficiency is achieved when 1 2 the resulting algorithm is mass conservative and naturally allows for the simulation of flooding and drying of tidal flats

semi implicit semi lagrangian modelling for environmental problems on staggered cartesian grids with cut cells in this paper we propose an extension of semi implicit models for the shallow water or euler equations on staggered cartesian meshes to handle the presence of cut cells at the boundaries of the computational domain semi lagrangian advection algorithms are also extended to staggered cartesian meshes with cut boundary cells a linear reconstruction algorithm and radial basis function interpolation are used respectively to compute approximate trajectories and to recover an accurate velocity field in cells intersected by the boundary the effective accuracy of radial basis function interpolators for semi lagrangian algorithms is assessed by simple numerical experiments timestep control techniques in the computation of the semi lagrangian trajectories are also discussed the accuracy of the resulting models for environmental flows is demonstrated by a number of test cases simulating nonlinear open channel flows and baroclinic flows over an isolated obstacle

a high order semi lagrangian discontinuous galerkin method for vlasov poisson simulations without operator splitting abstract in this paper we develop a high order semi lagrangian sl discontinuous galerkin dg method for nonlinear vlasov poisson vp simulations without operator splitting in particular we combine two recently developed novel techniques one is the high order non splitting sldg transport method cai et al 2017 4 and the other is the high order characteristics tracing technique proposed in qiu and russo 2017 29 the proposed method with up to third order accuracy in both space and time is locally mass conservative free of splitting error positivity preserving stable and robust for large time stepping size the sldg vp solver is applied to classic benchmark test problems such as landau damping and two stream instabilities for vp simulations efficiency and effectiveness of the proposed scheme is extensively tested tremendous cpu savings are shown by comparisons between the proposed sl dg scheme and the classical runge kutta dg method

spatial spline regression models summary we describe a model for the analysis of data distributed over irregularly shaped spatial domains with complex boundaries strong concavities and interior holes adopting an approach that is typical of functional data analysis we propose a spatial spline regression model that is computationally efficient allows for spatially distributed covariate information and can impose various conditions over the boundaries of the domain accurate surface estimation is achieved by the use of piecewise linear and quadratic finite elements

lagrange galerkin methods on spherical geodesic grids the shallow water equations abstract the weak lagrange galerkin finite element method for the 2d shallow water equations on the sphere is presented this method offers stable and accurate solutions because the equations are integrated along the characteristics the equations are written in 3d cartesian conservation form and the domains are discretized using linear triangular elements the use of linear triangular elements permits the construction of accurate by virtue of the second order spatial and temporal accuracies of the scheme and efficient by virtue of the less stringent cfl condition of lagrangian methods schemes on unstructured domains using linear triangles in 3d cartesian space allows for the explicit construction of area coordinate basis functions thereby simplifying the calculation of the finite element integrals the triangular grids are constructed by a generalization of the icosahedral grids that have been typically used in recent papers an efficient searching strategy for the departure points is also presented for these generalized icosahedral grids which involves very few floating point operations in addition a high order scheme for computing the characteristic curves in 3d cartesian space is presented a general family of runge kutta schemes results for six test cases are reported in order to confirm the accuracy of the scheme

a lagrange multiplier method for the finite element solution of elliptic interface problems using non matching meshes in this paper we propose a lagrange multiplier method for the finite element solution of multi domain elliptic partial differential equations using non matching meshes the interface lagrange multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes e g we will consider global polynomials as multiplier the ideas are illustrated using poisson s equation as a model and the proposed method is shown to be stable and optimally convergent numerical experiments demonstrating the theoretical results are also presented

mesh generation in curvilinear domains using high order elements the ability to construct suitable computational meshes is currently a significant limiting factor in the development of compact high order algorithms in very complex geometries compact high order algorithms such as spectral element or p type finite element techniques offer the potential of high accuracy if the solution is smooth and a well behaved mapping exists between the local sub domains and a standard region in this paper we address the problem of coupling high order algorithms with unstructured mesh generation and cad representation techniques we present three complementary strategies to alleviate the problem of generation of elemental regions with singular elemental mappings the first strategy investigates the influence of an anisometric parametric surface representation on the quality of high order unstructured meshes we show that a straightforward splitting of a standard linear mesh for use in a high order method can lead to the generation of distorted possibly invalid meshes even for simple computational domains such as cubes the second strategy uses hybrid meshes applying prismatic elements near curvilinear boundaries in the third and final strategy we investigate the use of curvature driven surface discretization to incorporate higher order information about the surface into the mesh generation using the computational reconstruction of the arterial bypass graft as an example computational domain we demonstrate how the combination of all three strategies leads to a valid high order discretization of the computational domain copyright 2001 john wiley sons ltd

a fast high resolution algorithm for linear convection problems particle transport method the paper is devoted to a novel explicit technique the particle transport method ptm for solving linear convection problems while being a lagrangian characteristic based method ptm has the advantage of eulerian methods to represent the solution on a fixed mesh the proposed approach belongs to the class of monotone high resolution numerical schemes possesses the property of unconditional stability and works with structured and unstructured meshes it is also demonstrated that the method has a linear computational complexity the performance of the presented algorithm is tested on one and two dimensional benchmark problems the numerical results confirm that the method has the 2nd order spatial accuracy and can be significantly faster than the grid based methods of the same order copyright 2006 john wiley sons ltd

an a priori error analysis of the local discontinuous galerkin method for elliptic problems in this paper we present the first a priori error analysis for the local discontinuous galerkin ldg method for a model elliptic problem for arbitrary meshes with hanging nodes and elements of various shapes we show that for stabilization parameters of order one the l2 norm of the gradient and the l2 norm of the potential are of order k and k 1 2 respectively when polynomials of total degree at least k are used if stabilization parameters of order h 1 are taken the order of convergence of the potential increases to k 1 the optimality of these theoretical results is tested in a series of numerical experiments on two dimensional domains

semi lagrangian methods for parabolic problems in divergence form semi lagrangian methods have traditionally been developed in the framework of hyperbolic equations but several extensions of the semi lagrangian approach to diffusion and advection diffusion problems have been proposed recently these extensions are mostly based on probabilistic arguments and share the common feature of treating second order operators in trace form which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics we propose here some basic ideas for treating second order operators in divergence form a general framework for constructing consistent schemes in one space dimension is presented and a specific case of nonconservative discretization is discussed in detail and analyzed finally an extension to possibly nonlinear problems in an arbitrary number of dimensions is proposed although the resulting discretization approach is only of first order in time numerical results in a number of test

high resolution flux based level set method a new high resolution flux based finite volume method for general advection equations in nondivergent form including a level set equation for moving interfaces is introduced the method is applicable to the case of nondivergence free velocity and to general unstructured grids in higher dimensions we show that the method is consistent and that the numerical solution fulfills the discrete minimum maximum principle numerical experiments show its second order accuracy for smooth solutions as well as for solutions with discontinuous derivatives and on general unstructured meshes numerical examples for passive transport and shrinking of dynamic interfaces including examples with topological changes are presented using locally adapted two dimensional and three dimensional grids

hp version discontinuous galerkin finite element method for semilinear parabolic problems we consider the hp version discontinuous galerkin finite element method hp dgfem with interior penalty for semilinear parabolic equations with locally lipschitz continuous nonlinearity subject to mixed nonhomogeneous dirichlet nonhomogeneous neumann boundary conditions our main concern is the error analysis of the spatially semidiscrete hp dgfem on shape regular spatial meshes we derive error bounds under various hypotheses on the regularity of the solution for both the symmetric and nonsymmetric versions of dgfem

an efficient characteristic galerkin scheme for the advection equation in 3 d conventional galerkin based finite element algorithms perform poorly when modeling advective transport here we develop and test a characteristic galerkin scheme to solve the unsteady three dimensional advective transport equation the algorithm uses tracking of the gaussian quadrature points to project the information from the eulerian background grid to the lagrangian grid numerical experiments were carried out to investigate the performance of this scheme despite speculations to the contrary in the literature this scheme does not suffer from instability problems further it is conservative and shows good phase characteristics with slight numerical diffusion we conclude that the characteristic galerkin method is a viable and efficient scheme for solving advection problems in three dimensions

liouvillian solutions of n th order homogeneous linear differential equations be a linear differential operator with coefficients in f a finite algebraic extension of q x we shall show that one can find in a finite number of steps a basis for the vector space of liouvillian solutions of l y 0 i e those solutions which can be built up from the rational functions by algebraic operations taking exponentials and by integration see section 2 for a precise definition in particular we show how to decide if all solution of l y 0 are liouvillian or if there are any solutions which are liouvillian our algorithm in conjunction with the algorithm in 8 also allows one to determine the algebraic relationships among the liouvillian solutions of l y 0 and in particular determine if all solutions are algebraic c f 111 for second order linear homogeneous equations over q x the problem of determining if all solutions are algebraic functions was considered by fuchs klein and schwarz but none of these mathematicians seems to have presented a complete decision procedure building on the work of klein baldassarri and dwork 21 have given such a procedure baldassarri 11 has extended this to consider linear homogeneous equations whose coefficients are algebraic functions for third order equations painleve and boulanger gave a procedure which in effect reduced the problem of finding algebraic solutions of l y 0 to the problem of effectively bounding the torsion of the jacobian variety of a given curve a complete procedure for deciding if all solutions of an n th order homo

the inverse distance weighted interpolation applied to a particular form of the path tubes method a simplified version of the inverse distance weighted interpolation formula was mathematically analyzed and employed in the context of a semi lagrangian scheme for the equation of advection the scheme used is a particular formulation of path tubes method a physically intuitive conservative method whose formulation is based on the theoretical foundations of the mechanics of continuous media which uses the so called reynolds s transport theorem to establish judiciously its main property written on the basis of a conservative integral equation the resulting algorithm is a five point explicit semi lagrangian scheme for incompressible flow from rigorous deductions based on the von neumann stability analysis it was proven that this explicit method is unconditionally stable the present algorithm was evaluated using test problems that are prototypes of more sophisticated mathematical models commonly employed in the prediction of intensity variations of weather fronts in addition our methodology was compared against two other methods available in the literature the proposed explicit semi lagrangian scheme was able to work with long time steps and proved to be accurate non oscillatory and non diffusive

high order discontinuous element based schemes for the inviscid shallow water equations spectral multidomain penalty and discontinuous galerkin methods two commonly used types of high order accuracyelement based schemes collocationbased spectral multidomain penalty methods smpm and nodal discontinuous galerkin methods dgm are compared in the framework of the inviscid shallow water equations differences and similarities in formulation are identified with the primary difference being the dissipative term in the rusanov form of the numerical flux for the dgm that provides additional numerical stability however it should be emphasized that to arrive at this equivalence between smpm and dgm requires making specific choices in the construction of both methods these choices are addressed in general both methods offer a multitude of choices in the penalty terms used to introduce boundary conditions and stabilize the numerical solution the resulting specialized class of smpm and dgm are then applied to a suite of six commonly considered geophysical flow test cases three linear and three non linear we also include results for a classical continuous galerkin i e spectral element method for comparison both the analysis and numerical experiments show that the smpm and dgm are essentially identical both methods can be shown to be equivalent for very special choices of quadrature rules and riemann solvers in the dgm along with special choices in the type of penalty term in the smpm although we only focus our studies on the inviscid shallow water equations the results presented should be applicable to other systems of nonlinear hyperbolic equations such as the compressible euler equations and extendable to the compressible and incompressible navier stokes equations where viscous terms are included

impact and importance of hyperdiffusion on the spectral element method a linear dispersion analysis abstract the spectral element method sem is a mimetic finite element method with several properties that make it a desirable choice for numerical modeling although the linear dispersion properties of this method have been analyzed extensively for the case of the 1d inviscid advection equation practical implementations of the sem frequently employ hyperdiffusion for stabilization as argued in this paper hyperdiffusion has a pronounced impact on the accuracy of the discrete wave modes and the dispersive properties of the sem when applied with an appropriately large coefficient hyperdiffusion is effective at removing the spectral gap and improving the stability of the 1d advection equation this study also considers the sem as applied to the 2d linearized shallow water equations where hyperdiffusion in the form of scalar diffusion divergence damping and vorticity damping are analyzed to the extent possible guidance on the choice of hyperdiffusion coefficients is provided a brief discussion of the comparative impact of local element filtering is included

multidimensional upwinding part ii decomposition of the euler equations into advection equations based on a genuine multidimensional numerical scheme called the method of transport we derive a form of the compressible euler equations capable of a linearization for any space dimension this form enables a rigorous error analysis of the linearization error without the knowledge of the numerical method used to solve the linear equations the generated error can be eliminated by special correction terms in the linear equations hence existing scalar high order methods can be used to solve the linear equations and obtain high order accuracy in space and time for the non linear conservation law in this approach the scalar version of the method of transport is used to solve the linear equations this method is multidimensional and reduces the solution of the partial differential equation to an integration process convergence histories presented at the end of the paper show that the numerical results agree with the theoretical predictions

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

a standard test set for numerical approximations to the shallow water equations in spherical geometry a suite of seven test cases is proposed for the evaluation of numerical methods intended for the solution of the shallow water equations in spherical geometry the shallow water equations exhibit the major difficulties associated with the horizontal dynamical aspects of atmospheric modeling on the spherical earth these cases are designed for use in the evaluation of numerical methods proposed for climate modeling and to identify the potential trade offs which must always be made in numerical modeling before a proposed scheme is applied to a full baroclinic atmospheric model it must perform well on these problems in comparison with other currently accepted numerical methods the cases are presented in order of complexity they consist of advection across the poles steady state geostrophically balanced flow of both global and local scales forced nonlinear advection of an isolated low zonal flow impinging on an isolated mountain rossby haurwitz waves and observed atmospheric states one of the cases is also identified as a computer performance algorithm efficiency benchmark for assessing the performance of algorithms adapted to massively parallel computers

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

robust untangling of curvilinear meshes this paper presents a technique that allows to untangle high order curvilinear meshes the technique makes use of unconstrained optimization where element jacobians are constrained to lie in a prescribed range through moving log barriers the untangling procedure starts from a possibly invalid curvilinear mesh and moves mesh vertices with the objective of producing elements that all have bounded jacobians bounds on jacobians are computed using the results of johnen et al 2012 2013 1 2 the technique is applicable to any kind of polynomial element for surface volume hybrid or boundary layer meshes a series of examples demonstrate both the robustness and the efficiency of the technique the final example involving a time explicit computation shows that it is possible to control the stable time step of the computation for curvilinear meshes through an alternative element deformation measure

lie theory applications to problems in mathematical finance and economics this paper is devoted to show and explain some applications of lie theory to solve some problems in economics and mathematical finance so we put forward and discuss mathematical aspects and approaches for several economic problems which have been previously considered in the literature besides we also show our advances on this topic mentioning some open problems for future research

energy conserving discontinuous galerkin spectral element method for the vlasov poisson system abstract we propose a new energy conserving spectral element discontinuous galerkin method for the approximation of the vlasov poisson system in arbitrary dimension using cartesian grids the method is derived from the one proposed in 4 with two modifications energy conservation is obtained by a suitable projection operator acting on the solution of the poisson problem rather than by solving multiple poisson problems and all the integrals appearing in the finite element formulation are approximated with gauss lobatto quadrature thereby yielding a spectral element formulation the resulting method has the following properties exact energy conservation up to errors introduced by the time discretization stability thanks to the use of upwind numerical fluxes high order accuracy and high locality for the time discretization we consider both runge kutta methods and exponential integrators and show results for 1d and 2d cases 2d and 4d in phase space respectively

evolution galerkin methods and their supraconvergence evolution galerkin methods for partial differential equations of the form u t l u 0 are characterised by i the use of some form of approximation to the corresponding evolution operator e t and ii projection onto an approximation space s h to obtain u n in this paper we concentrate on characteristic galerkin and lagrange galerkin methods to derive basic error estimates for multidimensional convection problems methods covered include those using recovery techniques to improve accuracy many schemes exhibit a supraconvergence phenomenon and a general technique for its analysis is given together with a number of particular examples

incremental remapping as a transport advection algorithm there are two fundamentally different strategies for solving the standard transport or continuity equation corresponding to whether it is expressed as a partial differential equation or as an integral statement of conservation the more common approach is to discretize the partial differential equation and to march the solution forward in time the alternative method is to project cell volumes along lagrangian trajectories as far forward or backward in time as desired and then to remap the resulting density distribution onto some target mesh this latter approach is known as remapping remapping has many advantages not the least of which is that the time step is limited only by accuracy considerations but it tends to be expensive and complex in this paper we show that if the time step is made sufficiently short such that trajectories are confined to the nearest neighbor cells then the remapping may be written as a flux form transport algorithm and it becomes nearly as simple and efficient as standard transport schemes the resulting method called incremental remapping retains most of the advantages of general remapping these include a geometric basis for transport b compatibility of associated tracer transport with simple tracer advection i e retention of tracer monotonicity properties and c efficient handling of multiple tracers since each additional tracer adds only a relatively small incremental cost

a semi lagrangian high order method for navier stokes equations we present a semi lagrangian method for advection diffusion and incompressible navier stokes equations the focus is on constructing stable schemes of second order temporal accuracy as this is a crucial element for the successful application of semi lagrangian methods to turbulence simulations we implement the method in the context of unstructured spectral hp element discretization which allows for efficient search interpolation procedures as well as for illumination of the nonmonotonic behavior of the temporal advection error of the form o tk xp 1 t we present numerical results that validate this error estimate for the advection diffusion equation and we document that such estimate is also valid for the navier stokes equations at moderate or high reynolds number two and three dimensional laminar and transitional flow simulations suggest that semi lagrangian schemes are more efficient than their eulerian counterparts for high order discretizations on nonuniform grids

classes of elementary function solutions to the cev model i abstract in the equity markets the stock price volatility increases as the stock price declines the classical black scholes merton bsm option pricing model does not reconcile with this association cox introduced the constant elasticity of variance cev model in 1975 in order to capture this inverse relationship between the stock price and its volatility an important parameter in the model is the parameter the elasticity of volatility the cev model subsumes some of the previous option pricing models for 0 1 2 and 1 the cev model reduces respectively to the bsm model the square root model of cox and ross and the bachelier model both in the case of the bsm model and in the case of the cev model it has become traditional to begin a discussion of option pricing by starting with the vanilla european calls and puts however there are simpler solutions to both models than those pertaining to the standard calls and puts mathematically it makes sense to investigate the simpler cases first in the case of bsm model simpler solutions are the log and power solutions similar simple solutions have not been studied so far for the cev model we use a group theoretic method kovacic s algorithm which has not been used before to problems of finance or economics and obtain new classes of elementary function solutions to the cev model for all half integer values of in particular when 5 2 2 3 2 1 1 3 2 2 5 2 we obtain four new classes of denumerably infinite elementary function solutions when 1 2 and 1 2 we obtain two new classes of denumerably infinite elementary function solutions whereas when 0 we find two elementary function solutions

numerical analysis of convection diffusion reaction problems with higher order characteristics finite elements part ii fully discretized scheme and quadrature formulas in this paper a higher order lagrange galerkin discretization method is analyzed when applied to a variable coefficient convection possibly degenerated diffusion reaction equation with mixed dirichlet robin boundary conditions in a previous paper a bermu dez m r nogueiras and c va zquez siam j numer anal to appear the proposed second order time discretization scheme has been rigorously introduced for exact and approximated characteristics moreover the l infty l 2 stability property and l infty l 2 error estimates of order o delta t 2 have been obtained as a continuation of that work consistency error estimates of order o delta t 2 h k are obtained for the fully discretized lagrange galerkin scheme moreover adequate quadrature formulas are proposed for the practical implementation of the method with particular finite element spaces finally some numerical tests illustrate the theoretical results and the performance of the combination of second order lagrange galerkin schemes with quadrature formulas

on the analysis of finite volume methods for evolutionary problems finite volume or finite element methods now dominate the modeling of steady flows because of their natural handling of complex configurations they should have a similar advantage for unsteady flows however their error analysis on nonuniform meshes has met a number of difficulties in this paper a new approach is adopted which makes greater use of a godunov formulation it opens the way for a fuller comparison of characteristic based methods with semidiscrete methods

mesh free semi lagrangian methods for transport on a sphere using radial basis functions abstract we present three new semi lagrangian methods based on radial basis function rbf interpolation for numerically simulating transport on a sphere the methods are mesh free and are formulated entirely in cartesian coordinates thus avoiding any irregular clustering of nodes at artificial boundaries on the sphere and naturally bypassing any apparent artificial singularities associated with surface based coordinate systems for problems involving tracer transport in a given velocity field the semi lagrangian framework allows these new methods to avoid the use of any stabilization terms such as hyperviscosity during time integration thus reducing the number of parameters that have to be tuned the three new methods are based on interpolation using 1 global rbfs 2 local rbf stencils and 3 rbf partition of unity for the latter two of these methods we find that it is crucial to include some low degree spherical harmonics in the interpolants standard test cases consisting of solid body rotation and deformational flow are used to compare and contrast the methods in terms of their accuracy efficiency conservation properties and dissipation dispersion errors for global rbfs spectral spatial convergence is observed for smooth solutions on quasi uniform nodes while high order accuracy is observed for the local rbf stencil and partition of unity approaches

on the relationship between semi lagrangian and lagrange galerkin schemes following a previous result stating their equivalence under constant advection speed semi lagrangian and lagrange galerkin schemes are compared in this paper in the situation of variable coefficient advection equations once known that semi lagrangian schemes can be proved to be equivalent to area weighted lagrange galerkin schemes via a suitable definition of the basis functions we will further prove that area weighted lagrange galerkin schemes represent a small more precisely an o delta t perturbation of exact lagrange galerkin schemes this equivalence implies a general result of stability for semi lagrangian schemes

a semi lagrangian discontinuous galerkin method for scalar advection by incompressible flows a new conservative semi lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form the approach combines the accuracy and conservation properties of the discontinuous galerkin dg method with the computational efficiency and robustness of semi lagrangian sl techniques unconditional stability in the von neumann sense is proved for the proposed discretization in the one dimensional case a monotonization technique is then introduced based on the flux corrected transport approach this yields a multi dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard dg methods the accuracy and stability of the method are further demonstrated by two dimensional tracer advection tests in the case of incompressible flows the comparison with results obtained by standard sl and dg methods highlights several advantages of the new technique

modified lagrange galerkin methods to integrate time dependent incompressible navier stokes equations a numerical study of modified lagrange galerkin mlg methods is presented for the incompressible navier stokes equations the convergence rate of second order in time methods is numerically proved in several test examples using both the minielement mathbb p 1 bubble mathbb p 1 and the mathbb p 2 mathbb p 1 taylor hood element a focus of this work is to ascertain the influence of numerical quadrature on the stability of the methods particularly when the reynolds number re is high all results presented in this study confirm that low order finite element mlg methods have the same accuracy as standard lagrange galerkin lg methods but the computational cost of solving navier stokes problems is notably reduced when mlg methods are used

upwind methods for the baernunziato equations and higher order reconstruction using artificial viscosity this article is devoted to the numerical discretisation of the hyperbolic two phase flow model of baer and nunziato a special attention is paid on the discretisation of intercell flux functions in the framework of finite volume and discontinuous galerkin approaches where care has to be taken to efficiently approximate the non conservative products inherent to the model equations various upwind approximate riemann solvers have been tested on a bench of discontinuous test cases new discretisation schemes are proposed in a discontinuous galerkin framework following the criterion of abgrall and the path conservative formalism a stabilisation technique based on artificial viscosity is applied to the high order discontinuous galerkin method and compared against classical tvd muscl finite volume flux reconstruction

a superconvergent linear fe approximation for the solution of an elliptic system of pdes the aim of this work is to study a nonstandard piecewise linear finite element method for elliptic systems of partial differential equations this nonstandard method was considered by the authors for scalar elliptic equations and for a planar elasticity problem the method enables us to compute a superconvergent numerical approximation to the solution of the system of partial differential equations

stability monotonicity maximum and minimum principles and implementation of the volume corrected characteristic method we consider the volume corrected characteristics mixed method vccmm for tracer transport problems the volume correction adjustment maintains the local volume conservation of bulk fluids and the numerical convergence of the method we discuss some details of implementation by considering the scheme from an algebraic point of view we show that the volume correction adjustment is important for stability and necessary for the monotonicity and the maximum and minimum principles of the method we also derive a relatively weaker stability property for the uncorrected characteristic mixed method cmm some numerical experiments of a quarter five spot pattern of wells are given to verify our theoretical results and compare the concentration errors of vccmm and cmm due to random perturbations set up in the computation of the algorithm more numerical tests including one related to long time nuclear waste storage are given to compare vccmm with cmm and godunov s method showing that vccmm exhibits no overshoots or undershoots and less numerical diffusion

convergence and nonlinear stability of the lagrange galerkin method for the navier stokes equations the lagrange galerkin method is a numerical technique for solving convection dominated diffusion problems based on combining a special discretisation of the lagrangian material derivative along particle trajectories with a galerkin finite element method we present optimal error estimates for the lagrange galerkin mixed finite element approximation of the navier stokes equations in a velocity pressure formulation the method is shown to be nonlinearly stable

a superconvergent ldg hybridizable galerkin method for second order elliptic problems we identify and study an ldg hybridizable galerkin method which is not an ldg method for second order elliptic problems in several space dimensions with remarkable convergence properties unlike all other known discontinuous galerkin methods using polynomials of degree k 0 for both the potential as well as the flux the order of convergence in l 2 of both unknowns is k 1 moreover both the approximate potential as well as its numerical trace superconverge in l 2 like norms to suitably chosen projections of the potential with order k 2 this allows the application of element by element postprocessing of the approximate solution which provides an approximation of the potential converging with order k 2 in l 2 the method can be thought to be in between the hybridized version of the raviart thomas and that of the brezzi douglas marini mixed methods

a characteristic finite element algorithm for time dependent 3 d advection dominated transport using unstructured grids abstract an algorithm based on operator splitting has been successfully implemented for solving unsteady advection dominated transport problems in 3 d specifically the general operator integration factor splitting method of maday et al is applied to the unsteady advection diffusion equation with source sink terms the algorithm incorporates a 3 d characteristic galerkin scheme to treat advection and a standard galerkin treatment of the diffusion and source sink terms up to third order operator splitting was implemented and validated against several analytical solutions the algorithm showed the expected error behaviour and good performance in modeling advection dominated transport problems the practical utility and effectiveness of the proposed numerical scheme was further demonstrated by solving the graetz nusselt problem i e high peclet number mass heat transport in a fully developed pipe flow

a fully semi lagrangian discretization for the 2d incompressible navier stokes equations in the vorticity streamfunction formulation abstract a numerical method for the two dimensional incompressible navier stokes equations in vorticity streamfunction form is proposed which employs semi lagrangian discretizations for both the advection and diffusion terms thus achieving unconditional stability without the need to solve linear systems beyond that required by the poisson solver for the reconstruction of the streamfunction a description of the discretization of dirichlet boundary conditions for the semi lagrangian approach to diffusion terms is also presented numerical experiments on classical benchmarks for incompressible flow in simple geometries validate the proposed method

multidimensional upwinding part i the method of transport for solving the euler equations the aim of this paper is to show a new approach towards the discretization of multidimensional conservation laws the idea of transport associated with the solution of a scalar equation is used for the convective part of the compressible euler equations a multidimensional wave structure is derived to model the acoustic part of this non linear system that allows infinitely many propagation directions in the numerical method this provides the basic knowledge to construct a numerical method that does not rely on riemann solvers a more general definition of the waves together with the concept of consistency enables the design of a number of effective genuinely multidimensional methods

imex hdg dg a coupled implicit hybridized discontinuous galerkin and explicit discontinuous galerkin approach for shallow water systems abstract we propose imex hdg dg schemes for planar and spherical shallow water systems of interest is subcritical flow where the speed of the gravity wave is faster than that of nonlinear advection in order to simulate these flows efficiently we split the governing system into a stiff part describing the gravity wave and a non stiff part associated with nonlinear advection the former is discretized implicitly with the hdg method while an explicit runge kutta dg discretization is employed for the latter the proposed imex hdg dg framework 1 facilitates high order solutions both in time and space 2 avoids overly small time step sizes 3 requires only one linear system solve per time stage 4 relative to dg generates smaller and sparser linear systems while promoting further parallelism and 5 suppresses the fast modes in the system with a large time step size numerical results for various test cases demonstrate that our methods are beneficial for applications where non stiff terms are accurately treated while stiff terms are less accurately handled

adaptive discontinuous evolution galerkin method for dry atmospheric flow we present a new adaptive genuinely multidimensional method within the framework of the discontinuous galerkin method the discontinuous evolution galerkin deg method couples a discontinuous galerkin formulation with approximate evolution operators the latter are constructed using the bicharacteristics of multidimensional hyperbolic systems such that all of the infinitely many directions of wave propagation are considered explicitly in order to take into account multiscale phenomena that typically appear in atmospheric flows nonlinear fluxes are split into a linear part governing the acoustic and gravitational waves and a nonlinear part that models advection time integration is realized by the imex type approximation using the semi implicit second order backward differentiation formula bdf2 moreover in order to approximate efficiently small scale phenomena adaptive mesh refinement using the space filling curves via the amatos function library is employed four standard meteorological test cases are used to validate the new discontinuous evolution galerkin method for dry atmospheric convection comparisons with the rusanov flux a standard one dimensional approximate riemann solver used for the flux integration demonstrate better stability and accuracy as well as the reliability of the new multidimensional deg method

convergence analysis for a class of high order semi lagrangian advection schemes the convergence properties of a class of high order semi lagrangian schemes for pure advection equations are studied here in the framework of the theory of viscosity solutions we review the general convergence results for discrete time approximation schemes belonging to that class and we prove some a priori estimates in l infty and l2 for the rate of convergence of fully discrete schemes we prove then that a careful coupling of time and space discretizations can allow large time steps in the numerical integration still preserving the accuracy of the solutions several examples of schemes and numerical tests are presented

lagrange galerkin methods on spherical geodesic grids the weak lagrange galerkin finite element method for the 2d shallow water equations on the sphere is presented this method offers stable and accurate solutions because the equations are integrated along the characteristics the equations are written in 3d cartesian conservation form and the domains are discretized using linear triangular elements the use of linear triangular elements permits the construction of accurate by virtue of the second order spatial and temporal accuracies of the scheme and efficient by virtue of the less stringent cfl condition of lagrangian methods schemes on unstructured domains using linear triangles in 3d cartesian space allows for the explicit construction of area coordinate basis functions thereby simplifying the calculation of the finite element integrals the triangular grids are constructed by a generalization of the icosahedral grids that have been typically used in recent papers an efficient searching strategy for the departure points is also presented for these generalized icosahedral grids which involves very few floating point operations in addition a high order scheme for computing the characteristic curves in 3d cartesian space is presented a general family of runge kutta schemes results for six test cases are reported in order to confirm the accuracy of the scheme

conservative multimoment transport along characteristics for discontinuous galerkin methods a set of algorithms based on characteristic discontinuous galerkin methods is presented for tracer transport on the sphere the algorithms are designed to reduce message passing interface communica

strong and weak lagrange galerkin spectral element methods for the shallow water equations the lagrange galerkin spectral element method for the two dimensional shallow water equations is presented the equations are written in conservation form and the domains are discretized using quadrilateral elements lagrangian methods integrate the governing equations along the characteristic curves thus being well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations two types of lagrange galerkin methods are presented the strong and weak formulations the strong form relies mainly on interpolation to achieve high accuracy while the weak form relies pri marily on integration lagrange galerkin schemes offer an increased efficiency by virtue of their less stringent cfl condition the use of quadrilateral elements permits the construction of spectral type finite element methods that exhibit exponential convergence as in the conventional spectral method yet they are constructed locally as in the finite element method this is the spectral element method in this paper we show how to fuse the lagrange galerkin methods with the spectral element method and present results for two standard test cases in order to compare and contrast these two hybrid schemes 2003 published by elsevier science ltd

high order splitting methods for the incompressible navier stokes equations abstract a new pressure formulation for splitting methods is developed that results in high order time accurate schemes for the solution of the incompressible navier stokes equations in particular improved pressure boundary conditions of high order in time are introduced that minimize the effect of erroneous numerical boundary layers induced by splitting methods a new family of stiffly stable schemes is employed in mixed explicit implicit time intgration rules these schemes exhibit much broader stability regions as compared to adams family schemes typically used in splitting methods their stability properties remain almost constant as the accuracy of the integration increases so that robust third or higher order time accurate schemes can readily be constructed that remain stable at relatively large cfl number the new schemes are implemented within the framework of spectral element discretizations in space so that flexibility and accuracy is guaranteed in the numerical experimentation a model stokes problem is studied in detail and several examples of navier stokes solutions of flows in complex geometries are reported comparison is made with the previously used first order in time spectral element splitting and non splitting e g uzawa schemes high order splitting spectral element methods combine accuracy in space and time and flexibility in geometry and thus can be very efficient in direct simulations of turbulent flows in complex geometries

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

a staggered semi implicit spectral discontinuous galerkin scheme for the shallow water equations a spatially arbitrary high order semi implicit spectral discontinuous galerkin dg scheme for the numerical solution of the shallow water equations on staggered control volumes is derived and discussed the free surface elevation and the momentum are expressed in terms of the same polynomials that are used as basis and as test functions each unknown is however defined on a different set of control volumes that are spatially staggered with respect to each other a semi implicit time integration yields a stable and efficient mass conservative algorithm the use of a staggered mesh has the advantage that after substitution of the momentum equations into the mass conservation equation only one single block penta diagonal system must be solved for the new free surface location which is a scalar quantity subsequently the new momentum components are directly obtained the staggered semi implicit approach makes the present dg scheme different from other published dg schemes the sparse linear system of the resulting discrete wave equation for the free surface can be conveniently solved by a matrix free gmres algorithm furthermore for the shallow water equations the proposed scheme can be written as a quadrature free method where all surface and volume integrals can be precomputed once and for all on the reference element and assembled into universal matrices and tensors for the special case n 0 the proposed method reduces to a classical semi implicit finite difference scheme the proposed semi implicit scheme is particularly well suited for low froude number flows the method is validated on some typical academic benchmark problems using polynomial degrees of up to n 20

a conservative discontinuous galerkin semi implicit formulation for the navier stokes equations in nonhydrostatic mesoscale modeling a discontinuous galerkin dg finite element formulation is proposed for the solution of the compressible navier stokes equations for a vertically stratified fluid which are of interest in mesoscale nonhydrostatic atmospheric modeling the resulting scheme naturally ensures conservation of mass momentum and energy a semi implicit time integration approach is adopted to improve the efficiency of the scheme with respect to the explicit runge kutta time integration strategies usually employed in the context of dg formulations a method is also presented to reformulate the resulting linear system as a pseudo helmholtz problem in doing this we obtain a dg discretization closely related to those proposed for the solution of elliptic problems and we show how to take advantage of the numerical integration rules required in all dg methods for the area and flux integrals to increase the efficiency of the solution algorithm the resulting numerical formulation is then validated on a collection of classical two dimensional test cases including density driven flows and mountain wave simulations the performance analysis shows that the semi implicit method is indeed superior to explicit methods and that the pseudo helmholtz formulation yields further efficiency improvements

lie symmetries and differential galois groups of linear equations for a linear ordinary differential equation the lie algebra of its infinitesimal lie symmetries is compared with its differential galois group for this purpose an algebraic formulation of lie symmetries is developed it turns out that there is no direct relation between the two above objects in connection with this a new algorithm for computing the lie symmetries of a linear ordinary differential equation is presented

a high order semi implicit discontinuous galerkin method for the two dimensional shallow water equations on staggered unstructured meshes a well balanced spatially arbitrary high order accurate semi implicit discontinuous galerkin scheme is presented for the numerical solution of the two dimensional shallow water equations on staggered unstructured non orthogonal grids the semi implicit method is derived in such a fashion that all relevant integrals can be precomputed and stored in a preprocessing stage so that the extension to curved isoparametric elements is natural and does not increase the computational effort of the simulation at runtime for p 0 the resulting scheme becomes a generalization of the classical semi implicit finite volume finite difference scheme of casulli and walters 2000 25 but with less conditions on the grid geometry the method proposed in this paper allows large time steps with respect to the surface wave speed gh and is thus particularly suitable for low froude number flows the approach is validated on some typical academic benchmark problems using polynomial degrees up to p 6

a positivity preserving high order semi lagrangian discontinuous galerkin scheme for the vlasov poisson equations the vlasov poisson equations describe the evolution of a collisionless plasma represented through a probability density function pdf that self interacts via an electrostatic force one of the main difficulties in numerically solving this system is the severe time step restriction that arises from parts of the pdf associated with moderate to large velocities the dominant approach in the plasma physics community for removing these time step restrictions is the so called particle in cell pic method which discretizes the distribution function into a set of macro particles while the electric field is represented on a mesh several alternatives to this approach exist including fully lagrangian fully eulerian and so called semi lagrangian methods the focus of this work is the semi lagrangian approach which begins with a grid based eulerian representation of both the pdf and the electric field then evolves the pdf via lagrangian dynamics and finally projects this evolved field back onto the original eulerian mesh in particular we develop in this work a method that discretizes the 1 1 vlasov poisson system via a high order discontinuous galerkin dg method in phase space and an operator split semi lagrangian method in time second order accuracy in time is relatively easy to achieve via strang operator splitting with additional work using higher order splitting and a higher order method of characteristics we also demonstrate how to push this scheme to fourth order accuracy in time we show how to resolve all of the lagrangian dynamics in such a way that mass is exactly conserved positivity is maintained and high order accuracy is achieved the poisson equation is solved to high order via the smallest stencil local discontinuous galerkin ldg approach we test the proposed scheme on several standard test cases

a class of implicit explicit two step runge kutta methods this work develops new implicit explicit time integrators based on two step runge kutta methods the class of schemes of interest is characterized by linear invariant preservation and high stage orders theoretical consistency stability and stiff convergence analyses are performed to reveal the excellent properties of these methods the new framework offers extreme flexibility in the construction of partitioned integrators since no coupling conditions are necessary practical schemes of orders three four and six are constructed and are used to solve several test problems numerical results confirm the theoretical findings

semi implicit formulations of the navier stokes equations application to nonhydrostatic atmospheric modeling we present semi implicit implicit explicit formulations of the compressible navier stokes equations nse for applications in nonhydrostatic atmospheric modeling the compressible nse in nonhydrostatic atmospheric modeling include buoyancy terms that require special handling if one wishes to extract the schur complement form of the linear implicit problem we present results for five different forms of the compressible nse and describe in detail how to formulate the semi implicit time integration method for these equations finally we compare all five equations and compare the semi implicit formulations of these equations both using the schur and no schur forms against an explicit runge kutta method our simulations show that if efficiency is the main criterion it matters which form of the governing equations you choose furthermore the semi implicit formulations are faster than the explicit runge kutta method for all the tests studied especially if the schur form is used while we have used the spectral element method for discretizing the spatial operators the semi implicit formulations that we derive are directly applicable to all other numerical methods we show results for our five semi implicit models for a variety of problems of interest in nonhydrostatic atmospheric modeling including inertia gravity waves density current i e kelvin helmholtz instabilities and mountain test cases the latter test case requires the implementation of nonreflecting boundary conditions therefore we show results for all five semi implicit models using the appropriate boundary conditions required in nonhydrostatic atmospheric modeling no flux reflecting and nonreflecting boundary conditions nrbcs it is shown that the nrbcs exert a strong impact on the accuracy and efficiency of the models

computation of the noncentral gamma distribution this paper deals with the computation of upper and lower tail probabilities of the noncentral gamma distribution and with the computation of the noncentrality parameter given the upper or lower tail probability the tail probabilities are computed in an efficient and numerically stable manner with a given relative accuracy even for small probabilities and for a wide range of parameters where other algorithms found in the statistical literature fail the basic ideas in this paper can also be applied to the noncentral chisquare beta f and t distributions