on multiscale methods in petrov galerkin formulation in this work we investigate the advantages of multiscale methods in petrov galerkin pg formulation in a general framework the framework is based on a localized orthogonal decomposition of a high dimensional solution space into a low dimensional multiscale space with good approximation properties and a high dimensional remainder space which only contains negligible fine scale information the multiscale space can then be used to obtain accurate galerkin approximations as a model problem we consider the poisson equation we prove that a petrov galerkin formulation does not suffer from a significant loss of accuracy and still preserve the convergence order of the original multiscale method we also prove inf sup stability of a pg continuous and a discontinuous galerkin finite element multiscale method furthermore we demonstrate that the petrov galerkin method can decrease the computational complexity significantly allowing for more efficient solution algorithms as another application of the framework we show how the petrov galerkin framework can be used to construct a locally mass conservative solver for two phase flow simulation that employs the buckley leverett equation to achieve this we couple a pg discontinuous galerkin finite element method with an upwind scheme for a hyperbolic conservation law

finite difference heterogeneous multi scale method for homogenization problems in this paper we propose a numerical method the finite difference heterogeneous multi scale method fd hmm for solving multi scale parabolic problems based on the framework introduced in commun math sci 1 1 87 the numerical method relies on the use of two different schemes for the original equation at different grid level which allows to give numerical results at a much lower cost than solving the original equations we describe the strategy for constructing such a method discuss generalization for cases with time dependency random correlated coefficients nonconservative form and implementation issues finally the new method is illustrated with several test examples

entropy stable high order discontinuous galerkin methods for ideal compressible mhd on structured meshes abstract we present a discontinuous galerkin dg scheme with suitable quadrature rules 15 for ideal compressible magnetohydrodynamic mhd equations on structural meshes the semi discrete scheme is analyzed to be entropy stable by using the symmetrizable version of the equations as introduced by godunov 32 the entropy stable dg framework with suitable quadrature rules 15 the entropy conservative flux in 14 inside each cell and the entropy dissipative approximate godunov type numerical flux at cell interfaces to make the scheme entropy stable the main difficulty in the generalization of the results in 15 is the appearance of the non conservative source terms added in the modified mhd model introduced by godunov 32 which do not exist in the general hyperbolic system studied in 15 special care must be taken to discretize these source terms adequately so that the resulting dg scheme satisfies entropy stability total variation diminishing bounded tvd tvb limiters and bound preserving limiters are applied to control spurious oscillations we demonstrate the accuracy and robustness of this new scheme on standard mhd examples

a multiwave approximate riemann solver for ideal mhd based on relaxation ii numerical implementation with 3 and 5 waves in the first part of this work bouchut et al j comput phys 108 7 41 2007 we introduced an approximate riemann solver for one dimensional ideal mhd derived from a relaxation system we gave sufficient conditions for the solver to satisfy discrete entropy inequalities and to preserve positivity of density and internal energy in this paper we consider the practical implementation and derive explicit wave speed estimates satisfying the stability conditions of bouchut et al j comput phys 108 7 41 2007 we present a 3 wave solver that well resolves fast waves and material contacts and a 5 wave solver that accurately resolves the cases when two eigenvalues coincide a full 7 wave solver which is highly accurate on all types of waves will be described in a follow up paper we test the solvers on one dimensional shock tube data and smooth shear waves

goal oriented adaptivity of mixed gmsfem for flows in heterogeneous media abstract we present an adaptive goal oriented framework with the mixed generalized multiscale finite element method gmsfem to numerically solve the flow problem in heterogeneous media in this paper three different error indicators are considered two goal oriented and one a standard adaptive method for the global error each is used to approximate a number of quantities of interest given by interior edge fluxes in a high contrast field following the setting of mixed gmsfem on each coarse neighborhood of the computational domain the local error indicator introduced here is computed by the product of the residual norms of the primal and dual variational equations over coarse neighborhoods numerical results demonstrate the efficiency of this goal oriented method in comparison to a multiscale implementation of the dual weighted residual method and a standard adaptive method

a linearity preserving nodal variation limiting algorithm for continuous galerkin discretization of ideal mhd equations abstract in this work a stabilized continuous galerkin cg method for magnetohydrodynamics mhd is presented ideal compressible inviscid mhd equations are discretized in space on unstructured meshes using piecewise linear or bilinear finite element bases to get a semi discrete scheme stabilization is then introduced to the semi discrete method in a strategy that follows the algebraic flux correction paradigm this involves adding some artificial diffusion to the high order semi discrete method and mass lumping in the time derivative term the result is a low order method that provides local extremum diminishing properties for hyperbolic systems the difference between the low order method and the high order method is scaled element wise using a limiter and added to the low order scheme the limiter is solution dependent and computed via an iterative linearity preserving nodal variation limiting strategy the stabilization also involves an optional consistent background high order dissipation that reduces phase errors the resulting stabilized scheme is a semi discrete method that can be applied to inviscid shock mhd problems and may be even extended to resistive and viscous mhd problems to satisfy the divergence free constraint of the mhd equations we add parabolic divergence cleaning to the system various time integration methods can be used to discretize the scheme in time we demonstrate the robustness of the scheme by solving several shock mhd problems

uncertainty propagation using infinite mixture of gaussian processes and variational bayesian inference uncertainty propagation in flow through porous media problems is a challenging problem this is due to the high dimensionality of the random property fields e g permeability and porosity as well as the computational complexity of the models that are involved the usual approach is to construct a surrogate response surface and then use it instead of the expensive model to carry out the uncertainty propagation task however the construction of the surrogate surface is hampered by various aspects such as the limited number of model evaluations that one can afford the curse of dimensionality multi variate responses with non trivial correlations potential localized features of the response and or discontinuities in this work we extend upon the concept of the multi output gaussian process mgp to effectively deal with all of these difficulties simultaneously this non trivial extension involves an infinite mixture of mgp s that is trained using variational bayesian inference prior to observing any data a dirichlet process is used to generate the components of the mgp mixture the bayesian nature of the model allows for the quantification of the uncertainties due to the limited number of simulations i e we can derive error bars for the statistics of interest the automatic detection of the mixture components by the variational inference algorithm is able to capture discontinuities and localized features without adhering to ad hoc constructions finally correlations between the components of multi variate responses are captured by the underlying mgp model in a natural way

convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients we propose a multiscale finite element method for solving second order elliptic equations with rapidly oscillating coefficients the main purpose is to design a numerical method which is capable of correctly capturing the large scale components of the solution on a coarse grid without accurately resolving all the small scale features in the solution this is accomplished by incorporating the local microstructures of the differential operator into the finite element base functions as a consequence the base functions are adapted to the local properties of the differential operator in this paper we provide a detailed convergence analysis of our method under the assumption that the oscillating coefficient is of two scales and is periodic in the fast scale while such a simplifying assumption is not required by our method it allows us to use homogenization theory to obtain a useful asymptotic solution structure the issue of boundary conditions for the base functions will be discussed our numerical experiments demonstrate convincingly that our multiscale method indeed converges to the correct solution independently of the small scale in the homogenization limit application of our method to problems with continuous scales is also considered

physical constraints preserving lagrangian finite volume schemes for one and two dimensional special relativistic hydrodynamics abstract this paper studies the physical constraints preserving pcp lagrangian finite volume schemes for one and two dimensional special relativistic hydrodynamic rhd equations first the pcp property i e preserving the positivity of the rest mass density and the pressure and the bound of the velocity is proved for the first order accurate lagrangian scheme with the hllc riemann solver and forward euler time discretization the key is that the intermediate states in the hllc riemann solver are shown to be admissible or pcp when the hllc wave speeds are estimated suitably then the higher order accurate schemes are proposed by using the high order accurate strong stability preserving ssp time discretizations and the scaling pcp limiter as well as the weno reconstruction finally several one and two dimensional numerical experiments are conducted to demonstrate the accuracy and the effectiveness of the pcp lagrangian schemes in solving the special rhd problems involving strong discontinuities or large lorentz factor or low rest mass density or low pressure etc

an entropic solver for ideal lagrangian magnetohydrodynamics in this paper we adapt to the ideal 1d lagrangian mhd equations a class of numerical schemes of order one in time and space presented in an earlier paper and applied to the gas dynamics system they use some properties of systems of conservation laws with zero entropy flux which describe fluid models invariant by galilean transformation and reversible for regular solutions these numerical schemes satisfy an entropy inequality under cfl conditions in the last section we describe a particular scheme for the mhd equations and show with some numerical applications its robustness and accuracy the generalization to full eulerian multidimensional mhd will be the subject of a forthcoming paper

a multiwave approximate riemann solver for ideal mhd based on relaxation i theoretical framework we present a relaxation system for ideal magnetohydrodynamics mhd that is an extension of the suliciu relaxation system for the euler equations of gas dynamics from it one can derive approximate riemann solvers with three or seven waves that generalize the hllc solver for gas dynamics under some subcharacteristic conditions the solvers satisfy discrete entropy inequalities and preserve positivity of density and internal energy the subcharacteristic conditions are nonlinear constraints on the relaxation parameters relating them to the initial states and the intermediate states of the approximate riemann solver itself the 7 wave version of the solver is able to resolve exactly all material and alfven isolated contact discontinuities practical considerations and numerical results will be provided in another paper

well balanced second order finite element approximation of the shallow water equations with friction this paper investigates the approximation of the shallow water equations with topography and friction using continuous finite elements a new second order parameter free well balanced and posit

a conservative local multiscale model reduction technique for stokes flows in heterogeneous perforated domains abstract in this paper we present a new multiscale model reduction technique for the stokes flows in heterogeneous perforated domains the challenge in the numerical simulations of this problem lies in the fact that the solution contains many multiscale features and requires a very fine mesh to resolve all details in order to efficiently compute the solutions some model reductions are necessary to obtain a reduced model we apply the generalized multiscale finite element approach which is a framework allowing systematic construction of reduced models based on this general framework we will first construct a local snapshot space which contains many possible multiscale features of the solution using the snapshot space and a local spectral problem we identify dominant modes in the snapshot space and use them as the multiscale basis functions our basis functions are constructed locally with non overlapping supports which enhances the sparsity of the resulting linear system in order to enforce the mass conservation we propose a hybridized technique and uses a lagrange multiplier to achieve mass conservation we will mathematically analyze the stability and the convergence of the proposed method in addition we will present some numerical examples to show the performance of the scheme we show that with a few basis functions per coarse region one can obtain a solution with excellent accuracy

analysis of the heterogeneous multiscale method for parabolic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

on positivity preserving high order discontinuous galerkin schemes for compressible navier stokes equations we construct a local lax friedrichs type positivity preserving flux for compressible navier stokes equations which can be easily extended to multiple dimensions for generic forms of equations of state shear stress tensor and heat flux with this positivity preserving flux any finite volume type schemes including discontinuous galerkin dg schemes with strong stability preserving runge kutta time discretizations satisfy a weak positivity property with a simple and efficient positivity preserving limiter high order explicit runge kutta dg schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy numerical tests suggest that the positivity preserving flux and the positivity preserving limiter do not induce excessive artificial viscosity and the high order positivity preserving dg schemes without other limiters can produce satisfying non oscillatory solutions when the nonlinear diffusion in compressible navier stokes equations is accurately resolved

monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws abstract using the theoretical framework of algebraic flux correction and invariant domain preserving schemes we introduce a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations nonlinear scalar conservation laws and hyperbolic systems in contrast to flux corrected transport fct algorithms that apply limited antidiffusive corrections to bound preserving low order solutions our new limiting strategy exploits the fact that these solutions can be expressed as convex combinations of bar states belonging to a convex invariant set of physically admissible solutions each antidiffusive flux is limited in a way which guarantees that the associated bar state remains in the invariant set and preserves appropriate local bounds there is no free parameter and no need for limit fluxes associated with the consistent mass matrix of time derivative term separately moreover the steady state limit of the nonlinear discrete problem is well defined and independent of the pseudo time step in the case study for the euler equations the components of the bar states are constrained sequentially to satisfy local maximum principles for the density velocity and specific total energy in addition to positivity preservation for the density and pressure the results of numerical experiments for standard test problems illustrate the ability of built in convex limiters to resolve steep fronts in a sharp and nonoscillatory manner

the group finite element formulation abstract the introduction of trial solutions for groups of variables within a finite element formulation is shown to provide a significant economy with no loss of accuracy for steady solutions of the two dimensional burgers equations with linear rectangular elements the application of the group finite element to the convective term produces a threefold reduction in execution time with a small improvement in accuracy operation count estimates indicate even larger economies in three dimensions or if higher order nonlinearities occur or if higher order shape functions are used applications of the group formulation are described for inviscid and viscous laminar and turbulent separated and unseparated and incompressible and compressible flows

localized orthogonal decomposition techniques for boundary value problems in this paper we propose a local orthogonal decomposition method lod for elliptic partial differential equations with inhomogeneous dirichlet and neumann boundary conditions for this purpose we present new boundary correctors which preserve the common convergence rates of the lod even if the boundary condition has a rapidly oscillating fine scale structure we prove a corresponding a priori error estimate and present numerical experiments we also demonstrate numerically that the method is reliable with respect to thin conductivity channels in the diffusion matrix accurate results are obtained without resolving these channels by the coarse grid and without using patches that contain the channels

the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains in this contribution we analyze a generalization of the heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains the method was originally introduced by e and engquist commun math sci 1 1 87 132 2003 for homogenization problems in fixed domains it is based on a standard finite element approach on the macroscale where the stiffness matrix is computed by solving local cell problems on the microscale a posteriori error estimates are derived in l 2 by reformulating the problem into a discrete two scale formulation see also ohlberger in multiscale model simul 4 1 88 114 2005 and using duality methods afterwards numerical experiments are given in order to numerically evaluate the efficiency of the error estimate

upscaling method for problems in perforated domains with non homogeneous boundary conditions on perforations using non local multi continuum method nlmc abstract in this paper we present an upscaling method for problems in perforated domains with non homogeneous boundary conditions on perforations our methodology is based on the recently developed non local multicontinuum method nlmc the main ingredient of the method is the construction of suitable local basis functions with the capability of capturing multiscale features and non local effects we will construct multiscale basis functions for the coarse regions and additional multiscale basis functions for perforations with the aim of handling non homogeneous boundary conditions on perforations we start with describing our method for the laplace equation and then extending the framework for the elasticity problem and parabolic equations the resulting upscaled model has minimal size and the solution has physical meaning on the coarse grid we will present numerical results 1 for steady and unsteady problems 2 for laplace and elastic operators and 3 for neumann and robin non homogeneous boundary conditions on perforations numerical results show that the proposed method can provide good accuracy and provide significant reduction on the degrees of freedoms

analysis of the heterogeneous multiscale method for elliptic homogenization problems the heterogeneous multiscale method hmm is applied to various parabolic problems with multiscale coefficients these problems can be either linear or nonlinear optimal estimates are proved for the error between the hmm solution and the homogenized solution

a provably positive discontinuous galerkin method for multidimensional ideal magnetohydrodynamics the density and pressure are positive physical quantities in magnetohydrodynamics mhd design of provably positivity preserving pp numerical schemes for ideal compressible mhd is highly desirab

advances in fiber and film flow abstract we give an account of recent advances in fiber and film flow in particular we address local and global solvability of the equations of fiber spinning semigroup theory for the linearized equations of film casting and eigenvalue asymptotics via birkhoff sequences in general we will present the salient points in the proofs of the various results without dwelling on technicalities we shall provide new perspectives for the underlying analytical issues and shed light on some physical consequences

stability and regularity results for a size structured population model in the present paper a nonlinear size structured population dynamical model with size and density dependent vital rate functions is considered the linearization about stationary solutions is analyzed by semigroup and spectral methods in particular the spectrally determined growth property of the linearized semigroup is derived from its long term regularity these analytical results make it possible to derive linear stability and instability results under biologically meaningful conditions on the vital rates the principal stability criteria are given in terms of a modified net reproduction rate

examples of computational approaches for elliptic possibly multiscale pdes with random inputs we overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness the specific equations manipulated are linear elliptic and arise in the context of multiscale problems but the purpose is more general on a set of prototypical situations we investigate two critical issues present in many settings variance reduction techniques to obtain sufficiently accurate results at a limited computational cost when solving pdes with random coefficients and finite element techniques that are sufficiently flexible to carry over to geometries with random fluctuations some elements of theoretical analysis and numerical analysis are briefly mentioned numerical experiments although simple provide convincing evidence of the efficiency of the approaches

positivity preserving finite difference weighted eno schemes with constrained transport for ideal magnetohydrodynamic equations in this paper we utilize the maximum principle preserving flux limiting technique originally designed for high order weighted essentially nonoscillatory weno methods for scalar hyperbolic conservation laws to develop a class of high order positivity preserving finite difference weno methods for the ideal magnetohydrodynamic equations our scheme under the constrained transport framework can achieve high order accuracy a discrete divergence free condition and positivity of the numerical solution simultaneously numerical examples in one two and three dimensions are provided to demonstrate the performance of the proposed method

positivity preserving analysis of numerical schemes for ideal magnetohydrodynamics numerical schemes provably preserving the positivity of density and pressure are highly desirable for ideal magnetohydrodynamics mhd but the rigorous positivity preserving pp analysis remains

on godunov type methods near low densities abstract when the energy of a flow is largely kinetic many conservative differencing schemes may fail by predicting non physical states with negative density or internal energy we describe as positively conservative the subclass of schemes that by contrast always generate physical solutions from physical data and show that the godunov method is positively conservative it is also shown that no scheme whose interface flux derives from a linearised riemann solution can be positively conservative classes of data that will bring about the failure of such schemes are described however the harten lax van leer hlle scheme is positively conservative under certain conditions on the numerical wavespeeds and this observation allows the linearised schemes to be rescued by modifying the wavespeeds employed

error control for the localized reduced basis multiscale method with adaptive on line enrichment in this contribution we consider localized robust and efficient a posteriori error estimation of the localized reduced basis multiscale lrbms method for parametric elliptic problems with possibly heterogeneous diffusion coefficient the numerical treatment of such parametric multiscale problems is characterized by a high computational complexity arising from the multiscale character of the underlying differential equation and the additional parameter dependence the lrbms method can be seen as a combination of numerical multiscale methods and model reduction using reduced basis rb methods to efficiently reduce the computational complexity with respect to the multiscale as well as the parametric aspect of the problem simultaneously in contrast to the classical residual based error estimators currently used in rb methods we are considering error estimators that are based on conservative flux reconstruction and provide an efficient and rigorous bound on the full error with respect to the weak solut

invariant domain preserving discretization independent schemes and convex limiting for hyperbolic systems abstract we introduce an approximation technique for nonlinear hyperbolic systems with sources that is invariant domain preserving the method is discretization independent provided elementary symmetry and skew symmetry properties are satisfied by the scheme the method is formally first order accurate in space then we introduce a series of higher order methods when these methods violate the invariant domain properties they are corrected by a limiting technique that we call convex limiting after limiting the resulting methods satisfy all the invariant domain properties that are imposed by the user see theorem 7 24 and is formally high order accurate the two key novelties are that i limiting is done by enforcing bounds on quasiconcave functionals ii the bounds that are enforced on the solution at each time step are necessarily satisfied by the low order approximation

a positive muscl hancock scheme for ideal magnetohydrodynamics we present a highly robust second order accurate scheme for the euler equations and the ideal mhd equations the scheme is of predictor corrector type with a muscl scheme following as a special case the crucial ingredients are an entropy stable approximate riemann solver and a new spatial reconstruction that ensures positivity of mass density and pressure for multidimensional mhd a new discrete form of the powell source terms is vital to ensure the stability properties the numerical examples show that the scheme has superior stability compared to standard schemes while maintaining accuracy in particular the method can handle very low values of pressure i e low plasma b or high mach numbers and low mass densities

estimation of local modeling error and goal oriented adaptive modeling of heterogeneous materials i error estimates and adaptive algorithms abstract a theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented these quantities may for example represent averaged stresses on the surface of inclusions or mollifications of pointwise stresses or displacements or in general local features of the fine scale solution characterized by continuous linear functionals these estimators are used to construct goal oriented adaptive procedures in which models of the microstructure are adapted to deliver local features to a preset level of accuracy algorithms for implementing these procedures are discussed and preliminary numerical results are given the analysis is restricted to linear static heterogeneous elastic materials

a staggered mesh algorithm using high order godunov fluxes to ensure solenoidal magnetic fields in magnetohydrodynamic simulations the equations of magnetohydrodynamics mhd have been formulated as a hyperbolic system of conservation laws in that form it becomes possible to use higher order godunov schemes for their solution this results in a robust and accurate solution strategy however the magnetic field also satisfies a constraint that requires its divergence to be zero at all times this is a property that cannot be guaranteed in the zone centered discretizations that are favored in godunov schemes without involving a divergence cleaning step in this paper we present a staggered mesh strategy which directly uses the properly upwinded fluxes that are provided by a godunov scheme the process of directly using the upwinded fluxes relies on a duality that exists between the fluxes obtained from a higher order godunov scheme and the electric fields in a plasma by exploiting this duality we have been able to construct a higher order godunov scheme that ensures that the magnetic field remains divergence free up to the computer s round off error we have even presented a variant of the basic algorithm that uses multidimensional features in the flow to design an upwinded strategy that aligns itself with the predominant upwinded direction in the flow we have devised several stringent test problems to show that the scheme works robustly and accurately in all situations in doing so we have shown that a scheme that involves a collocation of magnetic field variables that is different from the one traditionally favored in the design of higher order godunov schemes can nevertheless offer the same robust and accurate performance of higher order godunov schemes provided the properly upwinded fluxes from the godunov methodology are used in the scheme s construction

grad div stabilization and subgrid pressure models for the incompressible navier stokes equations in this paper the grad div stabilization for the incompressible navier stokes finite element approximations is considered from two different viewpoints i as a variational multiscale approach for the pressure subgrid modeling and ii as a stabilization procedure of least square type some new error estimates for the linearized problem with the grad div stabilization are proved with the help of norms induced by the pressure schur complement operator we discuss the stabilization parameter choice arising in the frameworks of least square and multiscale methods and consider assumptions which allow to relate both approaches

flux corrected transport algorithms for continuous galerkin methods based on high order bernstein finite elements abstract this work extends the flux corrected transport fct methodology to arbitrary order continuous finite element discretizations of scalar conservation laws on simplex meshes using bernstein polynomials as local basis functions we constrain the total variation of the numerical solution by imposing local discrete maximum principles on the bezier net the design of accuracy preserving fct schemes for high order bernstein bezier finite elements requires the development of new algorithms and or generalization of limiting techniques tailored for linear and multilinear lagrange elements in this paper we propose i a new discrete upwinding strategy leading to local extremum bounded low order approximations with compact stencils ii high order variational stabilization based on the difference between two gradient approximations and iii new localized limiting techniques for antidiffusive element contributions the optional use of a smoothness indicator based on a second derivative test makes it possible to potentially avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions the accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1d and 2d

a positivity preserving lagrangian discontinuous galerkin method for ideal magnetohydrodynamics equations in one dimension abstract in this paper we propose a conservative lagrangian discontinuous galerkin dg scheme for solving the ideal compressible magnetohydrodynamics mhd equations in one dimensional this scheme can preserve positivity of physically positive variables such as density and thermal pressure we first develop a lagrangian hlld approximate riemann solver which can keep positivity preserving property under some appropriate signal speeds with this solver a first order positivity preserving lagrangian dg scheme can be constructed then we design a high order positivity preserving and conservative lagrangian dg scheme by using the strong stability preserving ssp high order time discretizations and the positivity preserving scaling limiter we adopt tvb minmod limiter with the local characteristic fields for the ideal mhd system to control spurious oscillations around the shock wave some numerical examples are presented to demonstrate the accuracy and positivity preserving property of our scheme

a flux corrected transport algorithm for handling the close packing limit in dense suspensions convection of a scalar quantity by a compressible velocity field may give rise to unbounded solutions or nonphysical overshoots at the continuous and discrete level in this paper we are concerned with solving continuity equations that govern the evolution of volume fractions in eulerian models of disperse two phase flows an implicit galerkin finite element approximation is equipped with a flux limiter for the convective terms the fully multidimensional limiting strategy is based on a flux corrected transport fct algorithm this nonlinear high resolution scheme satisfies a discrete maximum principle for divergence free velocities and ensures positivity preservation for arbitrary velocity fields to enforce an upper bound that corresponds to the maximum packing limit an fct like overshoot limiter is applied to the converged convective fluxes at the end of each time step this postprocessing step imposes an additional physical constraint on the numerical solution to the unconstrained mathematical model numerical results for 2d implosion problems illustrate the performance of the proposed limiting procedure

a positive conservative method for magnetohydrodynamics based on hll and roe methods the exact riemann problem solutions of the usual equations of ideal magnetohydrodynamics mhd can have negative pressures if the initial data has b 0 this creates a problem for numerical solving because in a first order finite volume conservative godunov type method one cannot avoid jumps in the normal magnetic field component even if the magnetic field was divergenceless in the three dimensional sense we show that by allowing magnetic monopoles in mhd equations and properly taking into account the magnetostatic contribution to the lorentz force an additional source term appears in faraday s law only using the harten lax vanleer hll riemann solver and discretizing the source term in a specific manner we obtain a method which is positive and conservative we show positivity by extensive numerical experimentation this mhd hll method is positive and conservative but rather diffusive thus we show how to hybridize this method with the roe method to obtain a much higher accuracy while still retaining positivity the result is a fully robust positive conservative scheme for ideal mhd whose accuracy and efficiency properties are similar to the first order roe method and which keeps b small in the same sense as powell s method as a special case a method with similar characteristics for accuracy and robustness is obtained for the euler equations as well

ideal glm mhd about the entropy consistent nine wave magnetic field divergence diminishing ideal magnetohydrodynamics equations abstract the paper presents two contributions in the context of the numerical simulation of magnetized fluid dynamics first we show how to extend the ideal magnetohydrodynamics mhd equations with an inbuilt magnetic field divergence cleaning mechanism in such a way that the resulting model is consistent with the second law of thermodynamics as a byproduct of these derivations we show that not all of the commonly used divergence cleaning extensions of the ideal mhd equations are thermodynamically consistent secondly we present a numerical scheme obtained by constructing a specific finite volume discretization that is consistent with the discrete thermodynamic entropy it includes a mechanism to control the discrete divergence error of the magnetic field by construction and is galilean invariant we implement the new high order mhd solver in the adaptive mesh refinement code flash where we compare the divergence cleaning efficiency to the constrained transport solver available in flash unsplit staggered mesh scheme

positivity preserving dg and central dg methods for ideal mhd equations ideal mhd equations arise in many applications such as astrophysical plasmas and space physics and they consist of a system of nonlinear hyperbolic conservation laws the exact density r and pressure p should be non negative numerically such positivity property is not always satisfied by approximated solutions one can encounter this when simulating problems with low density high mach number or much large magnetic energy compared with internal energy when this occurs numerical instability may develop and the simulation can break down in this paper we propose positivity preserving discontinuous galerkin and central discontinuous galerkin methods for solving ideal mhd equations by following x zhang c w shu journal of computational physics 229 2010 8918 8934 in one dimension the positivity preserving property is established for both methods under a reasonable assumption the performance of the proposed methods in terms of accuracy stability and positivity preserving property is demonstrated through a set of one and two dimensional numerical experiments the proposed methods formally can be of any order of accuracy

residual driven online generalized multiscale finite element methods the construction of local reduced order models via multiscale basis functions has been an area of active research in this paper we propose online multiscale basis functions which are constructed using the offline space and the current residual online multiscale basis functions are constructed adaptively in some selected regions based on our error indicators we derive an error estimator which shows that one needs to have an offline space with certain properties to guarantee that additional online multiscale basis function will decrease the error this error decrease is independent of physical parameters such as the contrast and multiple scales in the problem the offline spaces are constructed using generalized multiscale finite element methods gmsfem we show that if one chooses a sufficient number of offline basis functions one can guarantee that additional online multiscale basis functions will reduce the error independent of contrast we note that the construction of online basis functions is motivated by the fact that the offline space construction does not take into account distant effects using the residual information we can incorporate the distant information provided the offline approximation satisfies certain properties in the paper theoretical and numerical results are presented our numerical results show that if the offline space is sufficiently large in terms of the dimension such that the coarse space contains all multiscale spectral basis functions that correspond to small eigenvalues then the error reduction by adding online multiscale basis function is independent of the contrast we discuss various ways computing online multiscale basis functions which include a use of small dimensional offline spaces

second order invariant domain preserving approximation of the euler equations using convex limiting a new second order method for approximating the compressible euler equations is introduced the method preserves all the known invariant domains of the euler system positivity of the density posi

provably positive high order schemes for ideal magnetohydrodynamics analysis on general meshes this paper proposes and analyzes arbitrarily high order discontinuous galerkin dg and finite volume methods which provably preserve the positivity of density and pressure for the ideal magnetohydrodynamics mhd on general meshes unified auxiliary theories are built for rigorously analyzing the positivity preserving pp property of numerical mhd schemes with a harten lax van leer hll type flux on polytopal meshes in any space dimension the main challenges overcome here include establishing certain relation between the pp property and a discrete divergence of magnetic field on general meshes and estimating proper wave speeds in the hll flux to ensure the pp property in the 1d case we prove that the standard dg and finite volume methods with the proposed hll flux are pp under a condition accessible by a pp limiter for the multidimensional conservative mhd system the standard dg methods with a pp limiter are not pp in general due to the effect of unavoidable divergence error in the magnetic field we construct provably pp high order dg and finite volume schemes by proper discretization of the symmetrizable mhd system with two divergence controlling techniques the locally divergence free elements and suitably discretized godunov powell source term the former technique leads to zero divergence within each cell while the latter controls the divergence error across cell interfaces our analysis reveals in theory that a coupling of these two techniques is very important for positivity preservation as they exactly contribute the discrete divergence terms which are absent in standard multidimensional dg schemes but crucial for ensuring the pp property several numerical tests further confirm the pp property and the effectiveness of the proposed pp schemes unlike the conservative mhd system the exact smooth solutions of the symmetrizable mhd system are proved to retain the positivity even if the divergence free condition is not satisfied our analysis and findings further the understanding at both discrete and continuous levels of the relation between the pp property and the divergence free constraint

mixed gmsfem for second order elliptic problem in perforated domains we consider a class of second order elliptic problems in perforated domains with homogeneous neumann boundary condition it is well known that numerically solving these problems require a very fine computational mesh and some model reduction techniques are therefore necessary we will develop a new model reduction technique based on the generalized multiscale finite element method gmsfem the gmsfem has been applied successfully to second order elliptic problems in perforated domains with dirichlet boundary conditions chung et al 2015 however due to the use of multiscale partition of unity functions the same method cannot be applied to the case with neumann boundary conditions the aim of this paper is to develop a new mixed gmsfem based on a piecewise constant approximation for pressure and a multiscale approximation for velocity giving a mass conservative method the method can handle the neumann boundary condition naturally the multiscale basis functions for velocity are constructed by some carefully chosen local snapshot spaces and local spectral decompositions the spectral convergence of the method is analyzed moreover by using some local error indicators the basis functions can be added locally and adaptively we also consider an online procedure for the construction of new basis functions in the online stage in order to capture the distant effects we will present some numerical examples to show the performance of the method

comparison of some flux corrected transport and total variation diminishing numerical schemes for hydrodynamic and magnetohydrodynamic problems two versions of flux corrected transport and two versions of total variation diminishing schemes are tested for several one and two dimensional hydrodynamic and magnetohydrodynamic problems two of the schemes ydfct and tvdlf are tested extensively for the first time the results give an insight into the limitations of the methods their relative strengths and weaknesses some subtle points of the algorithms and the effects of selecting different options for certain methods are emphasised

estimation of local modeling error and goal oriented adaptive modeling of heterogeneous materials part ii a computational environment for adaptive modeling of heterogeneous elastic solids this paper addresses a classical and largely unsolved problem given a structural component constructed of a heterogeneous elastic material that is in equilibrium under the action of applied loads determine local micromechanical features of its response e g local stresses and displacements in or around phase boundaries or in inclusions to an arbitrary preset level of accuracy it being understood that the microstructure is a priori unknown may be randomly distributed may exist at multiple spatial scales and may contain millions even billions of microscale components the approach described in this work begins with a mathematical abstraction of this problem in which the material body is modeled as an elastic solid with highly variable possibly randomly distributed elastic properties information on the actual character of the microstructure of given material bodies is determined by computerized tomography ct imaging a procedure is given for determining the effective material properties from imaging data using either deterministic or stochastic methods an algorithm is then described for determining local quantities of interest such as average stresses on inclusion boundaries to arbitrary accuracy relative to the fine scale model a new computational environment for implementing such analyses is presented which employs parallel adaptive hp finite element methods ct interfaces automatic meshing procedures and effectively adaptive modeling schemes within the basic premises on which the approach is based results of any specified accuracy can be obtained independently of the number of microscale components and constituents the results of several numerical experiments are presented

limiting and divergence cleaning for continuous finite element discretizations of the mhd equations abstract this work introduces a new type of constrained algebraic stabilization for continuous piecewise linear finite element approximations to the equations of ideal magnetohydrodynamics mhd at the first step of the proposed flux corrected transport fct algorithm the galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed at the second step limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field the limiting procedure developed for this stage is designed to enforce local maximum principles as well as positivity preservation for the density and thermodynamic pressure additionally it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints numerical studies for 2d test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal mhd benchmarks

embedded discontinuous galerkin transport schemes with localised limiters motivated by finite element spaces used for representation of temperature in the compatible finite element approach for numerical weather prediction we introduce locally bounded transport schemes for partially continuous finite element spaces the underlying high order transport scheme is constructed by injecting the partially continuous field into an embedding discontinuous finite element space applying a stable upwind discontinuous galerkin dg scheme and projecting back into the partially continuous space we call this an embedded dg transport scheme we prove that this scheme is stable in l 2 provided that the underlying upwind dg scheme is we then provide a framework for applying limiters for embedded dg transport schemes standard dg limiters are applied during the underlying dg scheme we introduce a new localised form of element based flux correction which we apply to limiting the projection back into the partially continuous space so that the whole transport scheme is bounded we provide details in the specific case of tensor product finite element spaces on wedge elements that are discontinuous p1 q1 in the horizontal and continuous p2 in the vertical the framework is illustrated with numerical tests

sequential limiting in continuous and discontinuous galerkin methods for the euler equations abstract we present a new predictor corrector approach to enforcing local maximum principles in piecewise linear finite element schemes for the compressible euler equations the new element based limiting strategy is suitable for continuous and discontinuous galerkin methods alike in contrast to synchronized limiting techniques for systems of conservation laws we constrain the density momentum and total energy in a sequential manner which guarantees positivity preservation for the pressure and internal energy after the density limiting step the total energy and momentum gradients are adjusted to incorporate the irreversible effect of density changes antidiffusive corrections to bounds compatible low order approximations are limited to satisfy inequality constraints for the specific total and kinetic energy an accuracy preserving smoothness indicator is introduced to gradually adjust lower bounds for the element based correction factors the employed smoothness criterion is based on a hessian determinant test for the density a numerical study is performed for test problems with smooth and discontinuous solutions

invariant domains and first order continuous finite element approximation for hyperbolic systems we propose a numerical method for solving general hyperbolic systems in any space dimension using forward euler time stepping and continuous finite elements on nonuniform grids the properties of the method are based on the introduction of an artificial dissipation that is defined so that any convex invariant set containing the initial data is an invariant domain for the method the invariant domain property is proved for any hyperbolic system provided a cfl condition holds the solution is also shown to satisfy a discrete entropy inequality for every admissible entropy of the system the method is formally first order accurate in space and can be made high order in time by using strong stability preserving algorithms this technique extends to continuous finite elements the work of d hoff math comp 33 1979 pp 1171 1193 d hoff trans amer math soc 289 1985 pp 591 610 and h frid arch ration mech anal 160 2001 pp 245 269

jeffreys fluids in forced elongation in this paper we study existence uniqueness and regularity of solutions for the equations governing the forced elongation of fluids with differential constitutive law of jeffreys type these equations consist of nonlinear first order hyperbolic equations in one spatial dimension forced elongation is imposed through velocity boundary conditions at the domain entry and exit the existence result is based on the schauder fixed point theorem and energy methods in the space of boundary regular functions

eigenvalue asymptotics in isothermal forced elongation in this paper we consider the formal linearization of the governing equations of forced elongation in the isothermal regime we give an asymptotic characterization of the spectrum of the associated strongly continuous semigroup to this end we develop rigorous asymptotic techniques to determine the asymptotic behavior of the eigenvalues by solving a characteristic integral equation in a generalized sector of the complex plane the spectral mapping theorem relating the spectrum of the semigroup to the spectrum of its generator proves essential numerical evidence will be given to demonstrate the accuracy of our asymptotic predictions

new directional vector limiters for discontinuous galerkin methods abstract second and higher order numerical approximations of conservation laws for vector fields call for the use of limiting techniques based on generalized monotonicity criteria in this paper we introduce a family of directional vertex based slope limiters for tensor valued gradients of formally second order accurate piecewise linear discontinuous galerkin dg discretizations the proposed methodology enforces local maximum principles for scalar products corresponding to projections of a vector field onto the unit vectors of a frame invariant orthogonal basis in particular we consider anisotropic limiters based on singular value decompositions and the gram schmidt orthogonalization procedure the proposed extension to hyperbolic systems features a sequential limiting strategy and a global invariant domain fix the pros and cons of different approaches to vector limiting are illustrated by the results of numerical studies for the two dimensional shallow water equations and for the euler equations of gas dynamics

an hllc type approximate riemann solver for ideal magnetohydrodynamics this paper presents a solver based on the hllc harten lax van leer contact wave approximate nonlinear riemann solver for gas dynamics for the ideal magnetohydrodynamics mhd equations written in conservation form it is shown how this solver also can be considered a modification of linde s adequate solver this approximation method is intended to resolve slow alfven and contact waves better than the original hll harten lax van leer solver compared to exact nonlinear solvers and roe s solver this new solver is computationally inexpensive in addition the method will exactly resolve isolated contacts and fast shocks the method also preserves positive density and pressure with two caveats first the numerical signal velocities the eigenvalues of the roe average matrix do not underestimate the physical signal velocities and second in a very few cases it may be required to change the wavespeeds of the riemann fan for the underlying hll method to guarantee positive pressures these conditi

runge kutta discontinuous galerkin methods for the special relativistic magnetohydrodynamics abstract this paper develops p k based non central and central runge kutta discontinuous galerkin dg methods with weno limiter for the one and two dimensional special relativistic magnetohydrodynamical rmhd equations k 1 2 3 the non central dg methods are locally divergence free while the central dg are exactly divergence free but have to find two approximate solutions defined on mutually dual meshes the adaptive weno limiter first identifies the troubled cells by using a modified tvb minmod function and then uses the weno technique and the cell average values of the dg solutions in the neighboring cells as well as the original cell averages of the troubled cells to locally reconstruct a new polynomial of degree 2 k 1 inside the troubled cells instead of the dg solution the weno limiting procedure does not destroy the locally or exactly divergence free property of magnetic field and is only employed for finite troubled cells so that the computational cost can be as little as possible several test problems in one and two dimensions are solved by using our non central and central runge kutta dg methods with weno limiter the numerical results demonstrate that our methods are stable accurate and robust in resolving complex wave structures

eigenvalue asymptotics in non isothermal elongational flow abstract we shall furnish an asymptotic description of the spectrum of a c0 semigroup generator arising in non isothermal elongational flow it will be shown that the spectrum coincides asymptotically with isolated eigenvalues lined up along a logarithmic curve the strategy for this result is based on delicate asymptotic estimates and ideas developed for the isothermal regime these estimates can be successfully employed only since the spectral mapping theorem is valid for the semigroup

an unsplit godunov method for ideal mhd via constrained transport we describe a single step second order accurate godunov scheme for ideal mhd based on combining the piecewise parabolic method ppm for performing spatial reconstruction the corner transport upwind ctu method of colella for multidimensional integration and the constrained transport ct algorithm for preserving the divergence free constraint on the magnetic field we adopt the most compact form of ct which requires the field be represented by area averages at cell faces we demonstrate that the fluxes of the area averaged field used by ct can be made consistent with the fluxes of the volume averaged field returned by a riemann solver if they obey certain simple relationships we use these relationships to derive new algorithms for constructing the ct fluxes at grid cell corners which reduce exactly to the equivalent one dimensional solver for plane parallel grid aligned flow we show that the ppm reconstruction algorithm must include multidimensional terms for mhd and we describe a number of important extensions that must be made to ctu in order for it to be used for mhd with ct we present the results of a variety of test problems to demonstrate the method is accurate and robust

an fct finite element scheme for ideal mhd equations in 1d and 2d abstract this paper presents an implicit finite element fe scheme for solving the equations of ideal magnetohydrodynamics in 1d and 2d the continuous galerkin approximation is constrained using a flux corrected transport fct algorithm the underlying low order scheme is constructed using a rusanov type artificial viscosity operator based on scalar dissipation proportional to the fast wave speed the accuracy of the low order solution can be improved using a shock detector which makes it possible to prelimit the added viscosity in a monotonicity preserving iterative manner at the fct correction step the changes of conserved quantities are limited in a way which guarantees positivity preservation for the density and thermal pressure divergence free magnetic fields are extracted using projections of the fct predictor into staggered finite element spaces forming exact sequences in the 2d case the magnetic field is projected into the space of raviart thomas finite elements numerical studies for standard test problems are performed to verify the ability of the proposed algorithms to enforce relevant constraints in applications to ideal mhd flows