runge kutta discontinuous galerkin methods with weno limiter for the special relativistic hydrodynamics this paper develops the p k based runge kutta discontinuous galerkin rkdg methods with weno limiter for the one and two dimensional special relativistic hydrodynamics k 1 2 3 which is an extension of the work j x qiu c w shu runge kutta discontinuous galerkin method using weno limiters siam j sci comput 26 2005 907 929 the weno limiter for the rkdg method is adaptively implemented via two following steps the troubled cells are first identified by using a tvb modified minmod function and then a new polynomial solution inside the troubled cells is locally reconstructed to replace the rkdg solution by using the weno technique based on the cell average values of the rkdg solution in the neighboring cells as well as the original cell averages of the troubled cells several test problems in one and two dimensions are computed using the developed rkdg methods with weno limiter the computations demonstrate that our methods are stable accurate and robust in maintaining accuracy for simulating flows in the special relativistic hydrodynamics

stepwise linear regression particular integrals for uncoupled thermoelasticity with boundary elements this paper presents a simple technique for the efficient treatment of non uniform body forces in boundary elements without the need for domain discretization the distribution of the nonuniform forces in the domain is described using regression polynomials particular integrals corresponding to these polynomials are developed for the solution of the non homogeneous differential equations of elasticity the method allows the use of low order polynomials to model a general distribution of thermal effects in the domain numerical data is given to provide comparisons of boundary only data versus combined boundary and domain data based regression polynomials

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

kinetic schemes for the relativistic gas dynamics a kinetic solution for the relativistic euler equations is presented this solution describes the flow of a perfect gas in terms of the particle density n the spatial part of the four velocity u and the inverse temperature in this paper we present a general framework for the kinetic scheme of relativistic euler equations which covers the whole range from the non relativistic limit to the ultra relativistic limit the main components of the kinetic scheme are described now i there are periods of free flight of duration m where the gas particles move according to the free kinetic transport equation ii at the maximization times tn n m the beginning of each of these free flight periods the gas particles are in local equilibrium which is described by juttners relativistic generalization of the classical maxwellian phase density iii at each new maximization time tn 0 we evaluate the so called continuity conditions which guarantee that the kinetic scheme satisfies the conservation laws and the entropy inequality these continuity conditions determine the new initial data at tn iv if in addition adiabatic boundary conditions are prescribed we can incorporate a natural reflection method into the kinetic scheme in order to solve the initial and boundary value problem in the limit m 0 we obtain the weak solutions of euler s equations including arbitrary shock interactions we also present a numerical shock reflection test which confirms the validity of our kinetic approach

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

differentiable monotonicity preserving schemes for discontinuous galerkin methods on arbitrary meshes this work is devoted to the design of interior penalty discontinuous galerkin dg schemes that preserve maximum principles at the discrete level for the steady transport and convection diffusion problems and the respective transient problems with implicit time integration monotonic schemes that combine explicit time stepping with dg space discretization are very common but the design of such schemes for implicit time stepping is rare and it had only been attained so far for 1d problems the proposed scheme is based on a piecewise linear dg discretization supplemented with an artificial diffusion that linearly depends on a shock detector that identifies the troublesome areas in order to define the new shock detector we have introduced the concept of discrete local extrema the diffusion operator is a graph laplacian instead of the more common finite element discretization of the laplacian operator which is essential to keep monotonicity on general meshes and in multi dimension the resulting nonlinear stabilization is non smooth and nonlinear solvers can fail to converge as a result we propose a smoothed twice differentiable version of the nonlinear stabilization which allows us to use newton with line search nonlinear solvers and dramatically improve nonlinear convergence a theoretical numerical analysis of the proposed schemes shows that they satisfy the desired monotonicity properties further the resulting operator is lipschitz continuous and there exists at least one solution of the discrete problem even in the non smooth version we provide a set of numerical results to support our findings

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

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

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

nonlinear analytic semiflows in this paper a local existence and regularity theory is given for nonlinear parabolic initial value problems x t f x t and quasilinear initial value problems x t a x t x t f x t this theory extends the theory of daprato and grisvard of 1979 and shows how various properties like analyticity of solutions can be derived as a direct corollary of the existence theorem

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 direct eulerian grp scheme for relativistic hydrodynamics two dimensional case this paper develops a direct eulerian generalized riemann problem grp scheme for two dimensional 2d relativistic hydrodynamics rhd it is an extension of the grp scheme for one dimensional 1d rhds z c yang p he h z tang j comput phys 230 2011 7964 7987 and the grp scheme for the non relativistic hydrodynamics m ben artzi j q li g warnecke j comput phys 218 2006 19 43 in order to derive the direct eulerian grp scheme the local grp of the split 2d rhd equations in the eulerian formulation has to be directly resolved by using corresponding riemann invariants and rankine hugoniot jump conditions so that the crucial and delicate lagrangian treatment in the original grp scheme m ben artzi j falcovitz j comput phys 55 1984 1 32 may be avoided an important difference of resolving the grp of the split 2d rhd equations from the grp of the 1d rhd equations or the non relativistic hydrodynamical equations is coming from the fact that the flow regions across the shock or rarefaction wave in the grp of the split 2d rhd equations are nonlinearly coupled through the lorentz factor which is also built in terms of the tangential velocities it is a purely multi dimensional relativistic feature several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed 2d grp scheme

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

flux correction tools for finite elements flux correction in the finite element context is addressed criteria for positivity of the numerical solution are formulated and the low order transport operator is constructed from the discrete high order operator by adding modulated dissipation so as to eliminate negative off diagonal entries the corresponding antidiffusive terms can be decomposed into a sum of genuine fluxes rather than element contributions which represent bilateral mass exchange between individual nodes thereby essentially one dimensional flux correction tools can be readily applied to multidimensional problems involving unstructured meshes the proposed methodology guarantees mass conservation and makes it possible to design both explicit and implicit fct schemes based on a unified limiting strategy numerical results for a number of benchmark problems illustrate the performance of the algorithm

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

two and three dimensional transient heat conduction and thermoelastic analyses by bem via efficient time convolution abstract efficient re integration based convolution algorithms are presented for a large class of problems involving transient two and three dimensional heat conduction and uncoupled thermoelastic analyses based on the earlier works of dargush and banerjee g f dargush p k banerjee application of boundary element method to transient heat conduction int j numer methods engrg 31 1991 1231 1248 g f dargush p k banerjee development of a boundary element method for time dependent planar thermoelasticity int j solids struct 25 1989 999 1021 the main motivation behind the current development is the fact that the storage based convolution approach used in earlier developments was not found to be very suitable for solving large scale practical problems because of substantial computer disk space requirement and the inability to handle time dependent boundary conditions in this paper a re integration based full convolution as well as an efficient and simplified re integration based fast convolution algorithms are presented in which all of the time convolution steps have been calculated by preserving the energy balance as expressed in the convolution while assuming equivalent values of variables to represent the total effect the accuracy and efficiency of the developed algorithms are demonstrated via numerical examples

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

notes on the eigensystem of magnetohydrodynamics the eigenstructure of the equations governing one dimensional ideal magnetohy drodynamics is examined motivated by the wish to exploit it for construction of high resolution computational algorithms the results are given in simple forms that avoid indeterminacy or degeneracy whenever possible the unavoidable indeterminacy near the magnetosonic or triple umbilic state is analysed and shown to cause no difficulty in evaluating a numerical flux function the structure of wave paths close to this singularity is obtained and simple expressions are presented for the structure coefficients that govern wave steepening

an iterative riemann solver for relativistic hydrodynamics an approximate method for solving the riemann problem is needed to construct godunov schemes for relativistic hydrodynamical equations such an approximate riemann solver is presented in this paper which treats all waves emanating from an initial discontinuity as themselves discontinuous therefore jump conditions for shocks are approximately used for rarefaction waves the solver is easy to implement in a godunov scheme and converges rapidly for relativistic hydrodynamics the fast convergence of the solver indicates the potential of a higher performance of a godunov scheme in which the solver is used

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

linearity preserving monotone local projection stabilization schemes for continuous finite elements abstract this paper presents some new tools for enforcing discrete maximum principles and or positivity preservation in continuous piecewise linear finite element approximations to convection dominated transport problems using a linear first order advection equation as a model problem we construct element level bilinear forms associated with first order artificial diffusion operators and their two scale counterparts the underlying design philosophy is similar to that behind local projection stabilization lps techniques and variational multiscale vms methods the difference lies in the structure of the local stabilization operator and in the way in which the resolved scales are detected the proposed stabilization term penalizes the difference between the nodal values and cell averages of the finite element solution in a manner which guarantees monotonicity and linearity preservation the value of the stabilization parameter is determined using a multidimensional limiter function designed to prevent unresolvable fine scale effects from creating undershoots or overshoots the result is a nonlinear high resolution scheme capable of resolving moving fronts and internal boundary layers as sharp localized nonoscillatory features the use of variational gradient recovery makes it possible to add high order background dissipation leading to improved approximation properties in smooth regions the numerical behavior of the constrained schemes is illustrated by a grid convergence study for stationary and time dependent test problems in two space dimensions

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

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

high order ader schemes for a unified first order hyperbolic formulation of newtonian continuum mechanics coupled with electro dynamics abstract in this paper we propose a new unified first order hyperbolic model of newtonian continuum mechanics coupled with electro dynamics the model is able to describe the behavior of moving elasto plastic dielectric solids as well as viscous and inviscid fluids in the presence of electro magnetic fields it is actually a very peculiar feature of the proposed pde system that viscous fluids are treated just as a special case of elasto plastic solids this is achieved by introducing a strain relaxation mechanism in the evolution equations of the distortion matrix a which in the case of purely elastic solids maps the current configuration to the reference configuration the model also contains a hyperbolic formulation of heat conduction as well as a dissipative source term in the evolution equations for the electric field given by ohm s law via formal asymptotic analysis we show that in the stiff limit the governing first order hyperbolic pde system with relaxation source terms tends asymptotically to the well known viscous and resistive magnetohydrodynamics mhd equations furthermore a rigorous derivation of the model from variational principles is presented together with the transformation of the euler lagrange differential equations associated with the underlying variational problem from lagrangian coordinates to eulerian coordinates in a fixed laboratory frame the present paper hence extends the unified first order hyperbolic model of newtonian continuum mechanics recently proposed in 110 42 to the more general case where the continuum is coupled with electro magnetic fields the governing pde system is symmetric hyperbolic and satisfies the first and second principle of thermodynamics hence it belongs to the so called class of symmetric hyperbolic thermodynamically compatible systems shtc which have been studied for the first time by godunov in 1961 61 and later in a series of papers by godunov and romenski 67 69 119 an important feature of the proposed model is that the propagation speeds of all physical processes including dissipative processes are finite the model is discretized using high order accurate ader discontinuous galerkin dg finite element schemes with a posteriori subcell finite volume limiter and using high order ader weno finite volume schemes we show numerical test problems that explore a rather large parameter space of the model ranging from ideal mhd viscous and resistive mhd over pure electro dynamics to moving dielectric elastic solids in a magnetic field

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

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 new bem formulation for tow and three dimensional elastoplasticity using particular integras new two and three dimensional bem formulations are developed for elastoplastic stress analysis these new procedures differ from previous work in that volume integration is not required to incorporate the nonlinear effects in the analysis instead initial stress rates are introduced in the bem system via particular integrals the present formulation is implemented in a general purpose multi region system and examples are presented to demonstrate the accuracy and versatility of the method

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

edge based nonlinear diffusion for finite element approximations of convection diffusion equations and its relation to algebraic flux correction schemes for the case of approximation of convection diffusion equations using piecewise affine continuous finite elements a new edge based nonlinear diffusion operator is proposed that makes the scheme satisfy a discrete maximum principle the diffusion operator is shown to be lipschitz continuous and linearity preserving using these properties we provide a full stability and error analysis which in the diffusion dominated regime shows existence uniqueness and optimal convergence then the algebraic flux correction method is recalled and we show that the present method can be interpreted as an algebraic flux correction method for a particular definition of the flux limiters the performance of the method is illustrated on some numerical test cases in two space dimensions

a cell centered lagrangian scheme for two dimensional compressible flow problems we present a new lagrangian cell centered scheme for two dimensional compressible flows the primary variables in this new scheme are cell centered i e density momentum and total energy are defined by their mean values in the cells the vertex velocities and the numerical fluxes through the cell interfaces are not computed independently contrary to standard approaches but are evaluated in a consistent manner due to an original solver located at the nodes the main new feature of the algorithm is the introduction of four pressures on each edge two for each node on each side of the edge this extra degree of freedom allows us to construct a nodal solver which fulfills two properties first the conservation of momentum and total energy is ensured second a semidiscrete entropy inequality is provided in the case of a one dimensional flow the solver reduces to the classical godunov acoustic solver it can be considered as its two dimensional generalization many numerical tests are presented they are representative test cases for compressible flows and demonstrate the robustness and the accuracy of this new solver

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

three dimensional transient thermo elastic analyses by the bie method the proposed method is the direct boundary integral equation bie method applied to three dimensional transient heat transfer analyses and to corresponding elastostatic analyses under thermal loading the mechanical and thermal problems are decoupled for those not familiar with the bie method a short survey of the basic principles is given to help them understand the mathematical treatment which is performed in the frame of distribution theory the tempered elementary solution of the transient heat transfer problem contains time and space variables the numerical treatment of the thermal equation is performed analytically with respect to time and numerically with respect to space in the same way as for steady state equations great care is paid to space integration in order to compute the values of the kernels precisely enough on the surface and inside the volume the computational treatments of both problems are realized in such a way as to minimize computing time differences between transient and steady state analyses on the one hand and between elastostatic analyses with and without thermal loading on the other are emphasized finally an analytical example shows the good behaviour of the proposed method to solve singular thermal problems and an industrial example shows the accuracy of the results obtained by using the bie method to solve three dimensional thermo elastic problems

riemann solver for relativistic hydrodynamics in this paper we construct an efficient accurate and rugged riemann solver for relativistic hydrodynamics the algorithm is an extension of the two shock approximation of colella to the relativistic regime the riemann solver constructed here is made to converge to the solution via iteration two different iterative techniques are presented one based on a secant method and the other on a newton method the method presented here provides an exact treatment of the transverse velocities across general oblique shocks this is a non trivial but very desirable property to have in a riemann solver for relativistic flow we also show the equivalence of our new formulation to the previous ones in the non relativistic limit

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

a new boundary element formulation for two and three dimensional thermoelasticity using particular integrals new two and three dimensional boundary element formulations are developed for steady state and transient uncoupled thermoelasticity these new procedures differ from previous work in that additional surface or volume integration is not required to incorporate thermal loads in the analysis instead thermal body forces are introduced in the boundary element system via particular integrals the present formulation is implemented in a general purpose multi region system and examples are presented to demonstrate the accuracy and versatility of the method

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

monotonicity preserving finite element schemes based on differentiable nonlinear stabilization in this work we propose a nonlinear stabilization technique for scalar conservation laws with implicit time stepping the method relies on an artificial diffusion method based on a graph laplacian operator it is nonlinear since it depends on a shock detector further the resulting method is linearity preserving the same shock detector is used to gradually lump the mass matrix the resulting method is led positivity preserving and also satisfies a global dmp lipschitz continuity has also been proved however the resulting scheme is highly nonlinear leading to very poor nonlinear convergence rates we propose a smooth version of the scheme which leads to twice differentiable nonlinear stabilization schemes it allows one to straightforwardly use newton s method and obtain quadratic convergence in the numerical experiments steady and transient linear transport and transient burgers equation have been considered in 2d using the newton method with a smooth version of the scheme we can reduce 10 to 20 times the number of iterations of anderson acceleration with the original non smooth scheme in any case these properties are only true for the converged solution but not for iterates in this sense we have also proposed the concept of projected nonlinear solvers where a projection step is performed at the end of every nonlinear iterations onto a fe space of admissible solutions the space of admissible solutions is the one that satisfies the desired monotonic properties maximum principle or positivity

the hele shaw problem and area preserving curve shortening motions we prove existence locally in time of a solution of the following hele shaw problem given a simply connected curve contained in a smooth bounded domain find the motion of the curve such that its normal velocity equals the jump of the normal derivatives of a function which is harmonic in the complement of the curve in and whose boundary value on the curve equals its curvature we show that this motion is a curve shortening motion which does not change the area of the region enclosed by the curve in case is the whole plane 2 we also show that if the initial curve is close to an equilibrium curve i e to a circle then there exists a global solution and the global solution tends to some circle exponentially fast as time tends to infinity

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

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

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

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

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

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

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

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

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

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

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

local bounds preserving stabilization for continuous galerkin discretization of hyperbolic systems abstract the objective of this paper is to present a local bounds preserving stabilized finite element scheme for hyperbolic systems on unstructured meshes based on continuous galerkin cg discretization in space a cg semi discrete scheme with low order artificial dissipation that satisfies the local extremum diminishing led condition for systems is used to discretize a system of conservation equations in space the low order artificial diffusion is based on approximate riemann solvers for hyperbolic conservation laws in this case we consider both rusanov and roe artificial diffusion operators in the rusanov case two designs are considered a nodal based diffusion operator and a local projection stabilization operator the result is a discretization that is led and has first order convergence behavior to achieve high resolution limited antidiffusion is added back to the semi discrete form where the limiter is constructed from a linearity preserving local projection stabilization operator the procedure follows the algebraic flux correction procedure usually used in flux corrected transport algorithms to further deal with phase errors or terracing common in fct type methods high order background dissipation is added to the antidiffusive correction the resulting stabilized semi discrete scheme can be discretized in time using a wide variety of time integrators numerical examples involving nonlinear scalar burgers equation and several shock hydrodynamics simulations for the euler system are considered to demonstrate the performance of the method for time discretization crank nicolson scheme and backward euler scheme are utilized

bound preserving discontinuous galerkin methods for relativistic hydrodynamics in this paper we develop a discontinuous galerkin dg method to solve the ideal special relativistic hydrodynamics rhd and design a bound preserving bp limiter for this scheme by extending the idea in x zhang and c w shu 2010 56 for rhd the density and pressure are positive and the velocity is bounded by the speed of light one difficulty in numerically solving the rhd in its conservative form is that the failure of preserving these physical bounds will result in ill posedness of the problem and blowup of the code especially in extreme relativistic cases the standard way in dealing with this difficulty is to add extra numerical dissipation while in doing so there is no guarantee of maintaining the high order of accuracy our bp limiter has the following features it can theoretically guarantee to preserve the physical bounds for the numerical solution and maintain its designed high order accuracy the limiter is local to the cell and hence is very easy to implement moreover it renders l 1 stability to the numerical scheme numerical experiments are performed to demonstrate the good performance of this bound preserving dg scheme even though we only discuss the bp limiter for dg schemes it can be applied to high order finite volume schemes such as weighted essentially non oscillatory weno finite volume schemes as well

a flux split algorithm applied to relativistic flows the equations of rfd can be written as a hyperbolic system of conservation laws by choosing an appropriate vector of unknowns we give an explicit formulation of the full spectral decomposition of the jacobian matrices associated with the fluxes in each spatial direction which is the essential ingredient of the techniques we propose in this paper these techniques are based on the recently derived flux formula of marquina a new way to compute the numerical flux at a cell interface which leads to a conservative upwind numerical scheme using the spectral decompositions in a fundamental way we construct high order versions of the basic first order scheme described by r donat and a marquina in j comput phys 125 42 1996 and test their performance in several standard simulations in one dimension two dimensional simulations include a wind tunnel with a flat faced step and a supersonic jet stream both of them in strongly ultrarelativistic regimes

a direct eulerian grp scheme for spherically symmetric general relativistic hydrodynamics the paper proposes a second order accurate direct eulerian generalized riemann problem grp scheme for the spherically symmetric general relativistic hydrodynamical rhd equations and a second order accurate discretization for the spherically symmetric einstein sse equations the former is directly using the riemann invariants and the runkine hugoniot jump conditions to analytically resolve the left and right nonlinear waves of the local grp in the eulerian formulation together with the local change of the metrics to obtain the limiting values of the time derivatives of the conservative variables along the cell interface and the numerical flux for the grp scheme while the latter utilizes the energy momentum tensor obtained in the grp solver to evaluate the fluid variables in the sse equations and keeps the continuity of the metrics at the cell interfaces several numerical experiments show that the grp scheme can achieve second order accuracy and high resolution and is effective for spherically symm

finite volume local evolution galerkin method for two dimensional relativistic hydrodynamics the paper proposes a second order accurate finite volume local evolution galerkin fvleg method for two dimensional special relativistic hydrodynamical rhd equations instead of using the dimensional splitting method or solving one dimensional local riemann problem in the direction normal to cell interface the fvleg method couples a finite volume formulation with the genuinely multi dimensional approximate local evolution operator which is derived by evolving the solutions of corresponding locally linearized rhd equations along all bicharacteristic directions several numerical examples are given to demonstrate the accuracy and the performance of the proposed fvleg method

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