fast high order ader schemes for linear hyperbolic equations a reformulation of the ader approach arbitrary high order schemes using derivatives for linear hyperbolic pde s is presented this reformulation leads to a drastic decrease of the computational effort a formula for the construction of ader schemes that are arbitrary high order accurate in space and time is given the accuracy for some selected schemes is shown numerically for the two dimensional linearized euler equations as a mathematical model for noise propagation in the time domain in aeroacoustics

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

duality for modules over finite rings and applications to coding theory this paper sets a foundation for the study of linear codes over finite rings the finite frobenius rings are singled out as the most appropriate for coding theoretic purposes because two classical theorems of mac williams the extension theorem and the mac williams identities generalize from finite fields to finite frobenius rings it is over frobenius rings that certain key identifications can be made between the ring and its complex characters

the time line interpolation method for large time step godunov type schemes abstract this paper describes the use of the time line interpolation procedure for the design of large time step godunov type schemes for systems of hyperbolic conservation laws in one dimension these schemes are based on a specific procedure to characterize the left and right states of the riemann problems at the cell interfaces when the courant number associated with the waves exceeds unity to do so the time line interpolation technique is used constant and linear reconstruction techniques are presented sonic or critical points are seen to be a source of difficulty in the algorithm and an appropriate treatment is proposed the algorithms are applied to the linear advection equation to the inviscid burgers equation and to the set of hyperbolic conservation laws that describe shallow water flow in one dimension these simulations show the superiority of the linear time reconstruction over the use of a constant time reconstruction when the linear reconstruction technique is used the modulus of the amplification factor of the scheme is equal to unity for all wave numbers inducing oscillations in the computed profile owing to the phase error the introduction of a slope limiter allows these oscillations to be eliminated but yields numerical diffusion thus restricting the range of applications of the scheme

decay estimates of solutions for the equations of motion of compressible viscous and heat conductive gases in an exterior domain in ℝ3 we consider the equations of motion of compressible viscous and heat conductive gases in an exterior domain in 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem

the generalized riemann problem for reactive flows abstract a generalized riemann problem is introduced for the equations of reactive non viscous compressible flow in one space dimension initial data are assumed to be linearly distributed on both sides of a jump discontinuity the resolution of the singularity is studied and the first order variation in time of flow variables is given in exact form

a time domain numerical modeling of two dimensional wave propagation in porous media with frequency dependent dynamic permeability an explicit finite difference scheme is presented for solving the two dimensional biot equations of poroelasticity across the full range of frequencies the key difficulty is to discretize the johnson koplik dashen jkd model which describes the viscous dissipations in the pores indeed the time domain version of biot jkd model involves order 1 2 fractional derivatives which amount to a time convolution product to avoid storing the past values of the solution a diffusive representation of fractional derivatives is used the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations the coefficients of the diffusive representation follow from an optimization procedure of the dispersion relation then various methods of scientific computing are applied the propagative part of the equations is discretized using a fourth order finite difference scheme whereas the diffusive part is solved exactly an immersed interface method is implemented to discretize the geometry on a cartesian grid and also to discretize the jump conditions at interfaces numerical experiments are proposed in various realistic configurations

a second order godunov type scheme for compressible fluid dynamics abstract a second order accurate scheme for the integration in time of the conservation laws of compressible fluid dynamics is presented two related versions are proposed one lagrangian and the second direct eulerian they both share the common ingredient which is a full analytic solution for the time derivatives of flow quantities at a jump discontinuity assuming initial nonvanishing slopes on both sides while this solution is an extension of the solution to the classical riemann problem the resulting schemes are second order extensions of godunov s methods in both cases they are very simple to implement in computer codes several numerical examples are shown where the only additional mechanism is a simple monotonicity algorithm

implementation of the grp scheme for computing radially symmetric compressible fluid flows the study of radially symmetric compressible fluid flows is interesting both from the theoretical and numerical points of view spherical explosion and implosion in air water and other media are well known problems in application typical difficulties lie in the treatment of singularity in the geometrical source and the imposition of boundary conditions at the symmetric center in addition to the resolution of classical discontinuities shocks and contact discontinuities in the present paper we present the implementation of direct generalized riemann problem grp scheme to resolve this issue the scheme is obtained directly by the time integration of the fluid flows our new contribution is to show rigorously that the singularity is removable and derive the updating formulae for mass and energy at the center together with the vanishing of the momentum we obtain new numerical boundary conditions at the center which are then incorporated into the grp scheme the main ingredient is the passage from the cartesian coordinates to the radially symmetric coordinates

time domain numerical modeling of wave propagation in 2d heterogeneous porous media this paper deals with the numerical modeling of wave propagation in porous media described by biot s theory the viscous efforts between the fluid and the elastic skeleton are assumed to be a linear function of the relative velocity which is valid in the low frequency range the coexistence of propagating fast compressional wave and shear wave and of a diffusive slow compressional wave makes numerical modeling tricky to avoid restrictions on the time step the biot s system is splitted into two parts the propagative part is discretized by a fourth order ader scheme while the diffusive part is solved analytically near the material interfaces a space time mesh refinement is implemented to capture the small spatial scales related to the slow compressional wave the jump conditions along the interfaces are discretized by an immersed interface method numerical experiments and comparisons with exact solutions confirm the accuracy of the numerical modeling the efficiency of the approach is illustrated by simulations of multiple scattering

code enumerators and tutte polynomials it is proved that the set of higher weight enumerators of a linear code over a finite field is equivalent to the tutte polynomial associated to the code an explicit expression for the tutte polynomial is given in terms of the subcode weights generalizations of these results are proved and are applied to codeword m tuples these general results are used to prove a very general macwilliams type identity for linear codes that generalizes most previous extensions of the macwilliams identity in addition a general and very useful matrix framework for manipulating weight and support enumerators of linear codes is presented

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

the pointwise estimates of diffusion wave for the navier stokes systems in odd multi dimensions we study the dissipation of solutions of the isentropic navier stokes equations in odd multi dimensions pointwise estimates of the time asymptotic shape of the solutions are obtained and shown to exhibit the generalized huygen s principle our approach is based on the detailed analysis of the green function of the linearized system this is used to study the coupling of nonlinear diffusion waves

on some decay properties of stokes semigroup of compressible viscous fluid flow in a 2 dimensional exterior domain in this paper we proved the local energy decay and some lp lq decay properties of solutions to the initial boundary value problem for the stokes equations of compressible viscous fluid flow in a 2 dimensional exterior domain kobayashi 1997 19 and kobayashi and shibata 1999 21 treated the same problem in a 3 dimensional exterior domain and our results obtained in this paper are an extension of results in kobayashi 1997 19 and kobayashi and shibata 1999 21 to the 2 dimensional case the fundamental solution to the resolvent equations for the stokes equations in r2 has a logarithmical singularity at 0 being a resolvent parameter while it is continuous up to 0 in r3 this difference requires us a new idea to prove the local energy decay estimate once getting the local energy decay estimate the required lp lq decay estimates in the exterior domain are obtained by combining the local energy estimate and the lp lq estimates in r2 by a cut off technique

optimal lp lq convergence rates for the compressible navier stokes equations with potential force in this paper we are concerned with the optimal lp lq convergence rates for the compressible navier stokes equations with a potential external force in the whole space under the smallness assumption on both the initial perturbation and the external force in some sobolev spaces the optimal convergence rates of the solution in lq norm with 2 q 6 and its first order derivative in l2 norm are obtained when the initial perturbation is bounded in lp with 1 p 6 5 the proof is based on the energy estimates on the solution to the nonlinear problem and some lp lq estimates on the semigroup generated by the corresponding linearized operator

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

application of the grp scheme to open channel flow equations the grp generalized riemann problem scheme originally conceived for gasdynamics is reformulated for the numerical integration of the shallow water equations in channels of rectangular cross section variable width and bed profile including a friction model for the fluid channel shear stress this scheme is a second order analytic extension of the first order godunov scheme based on time derivatives of flow variables at cell interfaces resulting from piecewise linear data reconstruction in cells the second order time integration is based on solutions to generalized riemann problems at cell interfaces thus accounting for the full governing equations including source terms the source term due to variable bed elevation is treated in a well balanced way so that quiescent flow is exactly replicated this is done by adopting the surface gradient method sgm several problems of steady or unsteady open channel flow are considered including the terms corresponding to variable channel width and bed elevation as well as to shear stress at the fluid channel interface using the manning friction model in all these examples remarkable agreement is obtained between the numerical integration and the exact or accurate solutions

on large time behavior of solutions to the compressible navier stokes equations in the half space in r3 the navier stokes equation for compressible viscous fluid is considered on the half space in r3 under the zero dirichlet boundary condition for the momentum with initial data near an arbitrarily given equilibrium of positive constant density and zero momentum time decay properties in l2 norms for solutions of the linearized problem are investigated to obtain the rate of convergence in l2 norms of solutions to the equilibrium when initial data are sufficiently close to the equilibrium in some lower bounds are derived for solutions to the linearized problem one of which indicates a nonlinear phenomenon not appearing in the case of the cauchy problem on the whole space

higher order multilevel framework for ader scheme in computational aeroacoustics abstract the versatile arbitrary derivative ader scheme is cast in a multilevel framework ml ader for fast solution of system of linear hyperbolic partial differential equations the solution is cycled through spatial operators of varying accuracy while retaining highest order accuracy by the use of a forcing function accuracy analysis of the multilevel framework including in the ml ader form is carried out in time domain as well as frequency domain results are obtained for benchmark problems in computational aeroacoustics at a much reduced computational cost

the generalized riemann problems for compressible fluid flows towards high order the generalized riemann problems grp for nonlinear hyperbolic systems of balance laws in one space dimension are now well known and can be formulated as follows given initial data which are smooth on two sides of a discontinuity determine the time evolution of the solution near the discontinuity in particular the grp of k 1 th order high resolution is based on an analytical evaluation of time derivatives up to kth order which turns out to be dependent only on the spatial derivatives up to kth order while the classical riemann problem serves as a primary building block in the construction of many numerical schemes most notably the godunov scheme the analytic study of grp will lead to an array of grp schemes which extend the godunov scheme currently there are extensive studies on the second order grp scheme which proves to be robust and is capable of resolving complex multidimensional fluid dynamic problems ben artzi and falcovitz 2003 4 in this paper we provide a new approach for solving the grp for the compressible flow system towards high order accuracy the derivation of second order grp solver is more concise compared to those in previous works and the third order grp or quadratic grp is resolved for the first time the latter is shown to be necessary through numerical experiments with strong discontinuities our method relies heavily on the new treatment of rarefaction waves indeed as a main technical step the propagation of singularities argument for the rarefaction fan is simplified by deriving the linear ode systems for the evolution of characteristic derivatives in the x t coordinates for generalized riemann invariants the case of sonic point is incorporated into a general treatment the accuracy of the derived grp solvers is justified and numerical examples are presented for the performance of the resulting schemes

rate of convergence of non stationary flow to the steady flow of compressible viscous fluid we consider a compressible viscous fluid affected by external forces of general form which are small and smooth enough in suitable norms in r 3 in shibata and tanaka y shibata k tanaka on the steady flow of compressible viscous fluid and its stability with respect to initial disturbance j math soc japan 55 2003 797 826 we proved the unique existence and some regularity of the steady flow and its globally in time stability with respect to a small initial disturbance in the h 3 framework in this paper we investigate the rate of the convergence of the non stationary flow to the corresponding steady flow when the initial data are small enough in the h 3 and also belong to l 6 5

asymptotic behavior of solutions of the compressible navier stokes equations on the half space asymptotic behavior of solutions of the compressible navier stokes equations on the half space r n n 2 is considered around a given constant equilibrium a solution formula for the linearized problem is derived and l p estimates for solutions of the linearized problem are obtained for 2 p it is shown that as in the case of the cauchy problem the leading part of the solution of the linearized problem is decomposed into two parts one that behaves like diffusion waves and the other one purely diffusively there however are some aspects different from the cauchy problem especially in considering spatial derivatives it is also shown that the solution of the linearized problem approaches for large times the solution of the nonstationary stokes problem in some l p spaces and as a result a solution formula for the nonstationary stokes problem is obtained large time behavior of solutions of the nonlinear problem is then investigated in l p spaces for 2 p by applying the results on the linearized analysis and the weighted energy method the results indicate that there may be some nonlinear interaction phenomena not appearing in the cauchy problem

heterogeneous domain decomposition for computational aeroacoustics this paper presents a strategy to accelerate the direct simulation of aeroacoustic problems in terms of cpu time the key idea is to introduce a heterogeneous domain decomposition the whole computational domain is subdivided into smaller domains in each of these subdomains the equations the discretization the mesh and the time step may be different and are adapted to the local behavior of the solution to reduce the total number of elements we propose the use of high order methods here the class of arbitrary high order using derivatives finite volume schemes on structured meshes and arbitrary high order using derivatives discontinuous galerkin methods on unstructured meshes seem a good choice to us the coupling procedure is validated and numerical results for the interface transmission problem and the single airfoil gust response problem from 4th computational aeroacoustics workshop on benchmark problems cp 2004 212954 nasa 2004 are presented together with the acoustic scattering problem at a cylinder and at multiple objects the coupling approach proves to be especially efficient for the propagation of sound in large domains

junction generalized riemann problem for stiff hyperbolic balance laws in networks in this paper we design a new implicit solver for the junction generalized riemann problem j grp which is based on a recently proposed implicit method for solving the generalized riemann problem grp for systems of hyperbolic balance laws we use the new j grp solver to construct an ader scheme that is globally explicit locally implicit and with no theoretical accuracy barrier in both space and time the resulting ader scheme is able to deal with stiff source terms and can be applied to non linear systems of hyperbolic balance laws in domains consisting on networks of one dimensional sub domains in this paper we specifically apply the numerical techniques to networks of blood vessels we report on a test problem with exact solution for a simplified network of three vessels meeting at a single junction which is then used to carry out a systematic convergence rate study of the proposed high order numerical methods schemes up to fifth order of accuracy in space and time are implemented and tested we then show the ability of the ader scheme to deal with stiff sources through a numerical simulation in a network of vessels an application to a physical test problem consisting of a network of 37 compliant silicon tubes arteries and 21 junctions reveals that it is imperative to use high order methods at junctions in order to preserve the desired high order of accuracy in the full computational domain for example it is demonstrated that a second order method throughout gives comparable results to a method that is fourth order in the interior of the domain and first order at junctions we are concerned with stiff hyperbolic balance laws in networks an implicit solver for the junction generalized riemann problem is proposed ader schemes of arbitrary accuracy in space and time for networks are constructed convergence rates studies of schemes up to fifth order are carried out it is necessary to match the accuracy at junctions to that of the rest of the domain

a direct eulerian grp scheme for compressible fluid flows a direct eulerian generalized riemann problem grp scheme is derived for compressible fluid flows riemann invariants are introduced as the main ingredient to resolve the generalized riemann problem grp directly for the eulerian formulation the crucial auxiliary lagrangian scheme in the original grp scheme is not necessary in the present framework the delicate sonic cases can be easily treated and the extension to multidimensional cases is obtained using the dimensional splitting technique

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

second order extension in space and time for a 3d cell centered lagrangian scheme abstract high energy density physics hedp flows are multi material flows characterized by strong shock waves and large changes in the domain shape due to rarefaction waves numerical schemes based on the lagrangian formalism are good candidates to model this kind of flows since the computational grid follows the fluid motion this provides accurate results around the shocks as well as a natural tracking of multi material interfaces and free surfaces the work proposed here is in continuity with the work of maire and nkonga 2009 more precisely the aim of this article is to develop robust and accurate methods for the 3d extension of the eucclhyd scheme with a second order extension based on muscl monotonic upstream centered scheme for conservation laws and grp generalized riemann problem procedures particular care is taken to preserve the symmetries and the monotonicity of the solutions the scheme robustness and accuracy are assessed on numerous lagrangian test cases for which the 3d extensions are very challenging

real gas flows in a duct a numerical scheme is presented for the solution of the euler equations of compressible flow of a real gas in a single spatial coordinate this include flow in a duct of variable cross section as well as flow with cylindrical or spherical symmetry and can prove useful when testing codes for the two dimensional equations governing compressible flow of a real gas the scheme is applied with success to a problem involving the interaction of converging and diverging cylindrical shocks for four equations of state and to a problem involving the reflection of a converging shock

solution of the generalized riemann problem for advection reaction equations we present a method for solving the generalized riemann problem for partial differential equations of the advection reaction type the generalization of the riemann problem here is twofold firstly the governing equations include nonlinear advection as well as reaction terms and secondly the initial condition consists of two arbitrary but infinitely differentiable functions an assumption that is consistent with piecewise smooth solutions of hyperbolic conservation laws the solution procedure local and valid for sufficiently small times reduces the solution of the generalized riemann problem of the inhomogeneous nonlinear equations to that of solving a sequence of conventional riemann problems for homogeneous advection equations for spatial derivatives of the initial conditions we illustrate the approach via the model advection reaction equation the inhomogeneous burgers equation and the nonlinear shallow water equations with variable bed elevation

remark on the generalized riemann problem method for compressible fluid flows the generalized riemann problem grp method was proposed for compressible fluid flows based on the lagrangian formulation m ben artzi j falcovitz a second order godunov type scheme for compressible fluid dynamics j comput phys 55 1 1984 1 32 and a direct eulerian version was developed in m ben artzi j li g warnecke a direct eulerian grp scheme for compressible fluid flows j comput phys 28 2006 19 43 by using the concept of riemann invariants the central feature of the grp method is the resolution of centered rarefaction waves in this note we show how to use the concept of riemann invariants in order to resolve the rarefaction waves in the lagrangian coordinate system and result in the grp scheme

comparison of the generalized riemann solver and the gas kinetic scheme for inviscid compressible flow simulations the generalized riemann problem grp scheme for the euler equations and gas kinetic scheme gks for the boltzmann equation are two high resolution shock capturing schemes for fluid simulations the difference is that one is based on the characteristics of the inviscid euler equations and their wave interactions and the other is based on the particle transport and collisions the similarity between them is that both methods can use identical muscl type initial reconstructions around a cell interface and the spatial slopes on both sides of a cell interface involve in the gas evolution process and the construction of a time dependent flux function although both methods have been applied successfully to the inviscid compressible flow computations their performances have never been compared since both methods use the same initial reconstruction any difference is solely coming from different underlying mechanism in their flux evaluation therefore such a comparison is important to help us to understand the correspondence between physical modeling and numerical performances since grp is so faithfully solving the inviscid euler equations the comparison can be also used to show the validity of solving the euler equations itself the numerical comparison shows that the grp exhibits a slightly better computational efficiency and has comparable accuracy with gks for the euler solutions in 1d case but the gks is more robust than grp for the 2d high mach number flow simulations the gks is absent from the shock instability and converges to the steady state solutions faster than the grp the grp has carbuncle phenomena likes a cloud hanging over exact riemann solvers the grp and gks use different physical processes to describe the flow motion starting from a discontinuity one is based on the assumption of equilibrium state with infinite number of particle collisions and the other starts from the non equilibrium free transport process to evolve into an equilibrium one through particle collisions the different mechanism in the flux evaluation deviates their numerical performance through this study we may conclude scientifically that it may not be valid to use the euler equations as governing equations to construct numerical fluxes in a discretized space with limited cell resolution to adapt the navier stokes ns equations is not valid either because the ns equations describe the flow behavior on the hydrodynamic scale and have no any corresponding physics starting from a discontinuity this fact alludes to the consistency of the euler and navier stokes equations with the continuum assumption and the necessity of a direct modeling of the physical process in the discretized space in the construction of numerical scheme when modeling very high mach number flows the development of numerical algorithm is similar to the modeling process in deriving the governing equations but the control volume here cannot be shrunk to zero

fast estimation from above of the maximum wave speed in the riemann problem for the euler equations this paper is concerned with the construction of a fast algorithm for computing the maximum speed of propagation in the riemann solution for the euler system of gas dynamics with the co volume equation of state the novelty in the algorithm is that it stops when a guaranteed upper bound for the maximum speed is reached with a prescribed accuracy the convergence rate of the algorithm is cubic and the bound is guaranteed for gasses with the co volume equation of state and the heat capacity ratio in the range 1 5 3

numerical modeling of 1d transient poroelastic waves in the low frequency range propagation of transient mechanical waves in porous media is numerically investigated in 1d the framework is the linear biot model with frequency independent coefficients the coexistence of a propagating fast wave and a diffusive slow wave makes numerical modeling tricky a method combining three numerical tools is proposed a fourth order ader scheme with time splitting to deal with the time marching a space time mesh refinement to account for the small scale evolution of the slow wave and an interface method to enforce the jump conditions at interfaces comparisons with analytical solutions confirm the validity of this approach

thermodynamical effects and high resolution methods for compressible fluid flows one of the fundamental differences of compressible fluid flows from incompressible fluid flows is the involvement of thermodynamics this difference should be manifested in the design of numerical schemes unfortunately the role of entropy expressing irreversibility is often neglected even though the entropy inequality as a conceptual derivative is verified for some first order schemes in this paper we refine the grp solver to illustrate how the thermodynamical variation is integrated into the design of high resolution methods for compressible fluid flows and demonstrate numerically the importance of thermodynamic effects in the resolution of strong waves as a by product we show that the grp solver works for generic equations of state and is independent of technical arguments

riemann solvers for water hammer simulations by godunov method the water hammer phenomenon can be described by a 2 2 system of hyperbolic partial differential equations pdes numerical solution of these pdes using finite volume schemes is investigated herein the underlying concept of the godunov scheme is the riemann problem that must be solved to provide fluxes between the computational cells the presence of the kinetic terms in the momentum equation determines the existence of shock and rarefaction waves which influence the design of the riemann solver approximation of the expressions for the riemann invariants and jump relationships can be used to derive first and second order approximate non iterative solvers the first order approximate solver is almost 2000 times faster than the exact one but gives inaccurate predictions when the densities and celerities are low the second order approximate solver gives very accurate solutions and is 300 times faster than the exact iterative one detailed indications are provided in the appendices for the practical implementation of the riemann solvers described herein copyright 2000 john wiley sons ltd

derivative riemann solvers for systems of conservation laws and ader methods in this paper we first briefly review the semi analytical method e f toro v a titarev solution of the generalized riemann problem for advection reaction equations proc roy soc london 458 2018 2002 271 281 for solving the derivative riemann problem for systems of hyperbolic conservation laws with source terms next we generalize it to hyperbolic systems for which the riemann problem solution is not available as an application example we implement the new derivative riemann solver in the high order finite volume ader advection schemes we provide numerical examples for the compressible euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes

a reduction procedure for generalized riemann problems with application to nonlinear transmission lines generalized simple wave solutions to quasilinear hyperbolic nonhomogeneous systems of pdes are obtained through the differential constraint method these solutions prove to be flexible enough to solve generalized riemann problems where discontinuous initial data are involved within such a theoretical framework the governing model of nonlinear transmission lines is investigated throughout

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

high order cut discontinuous galerkin methods with local time stepping for acoustics we propose a method to solve the acoustic wave equation on an immersed domain using the hybridizable discontinuous galerkin method for spatial discretization and the arbitrary derivative method wit

a global existence result for the compressible navier stokes equations in the critical l p framework the present paper is dedicated to the global well posedness issue for the barotropic compressible navier stokes system in the whole space mathbb r d with d 2 we aim at extending the work by danchin inventiones mathematicae 141 3 579 614 2000 to a critical framework which is not related to the energy space for small perturbations of a stable equilibrium state in the sense of suitable lp type besov norms we establish the global existence as a consequence like for incompressible flows one may exhibit a class of large highly oscillating initial velocity fields for which global existence and uniqueness holds true in passing we obtain new estimates for the linearized and the paralinearized systems which may be of interest for future works on compressible flows

numerical solution to rarefaction or shock wave duct area change interaction conclusion it may be said that vt rj prescribed by eq 5 gives the best correlation with measurement for k and further substantiates the claim of so and hwang1 that the solution eqs 3 7 represents the best solution for self preserving turbulent round jets furthermore eqs 3 7 also represent the solutions for incompressible heated round jets if the turbulent prandtl number prt is assumed to be constant then eq 3 with the exponent r 2 i2 replaced by ri2pr vn2 becomes the solution of the temperature equation since eqs 1 and 2 are equally applicable for incompressible heated round jets eqs 8 11 also represent solutions for such flows therefore the present results are just as valid for nonisothermal round jets

finite volume transport schemes we analyze arbitrary order linear finite volume transport schemes and show asymptotic stability in l 1 and l for odd order schemes in dimension one it gives sharp fractional order estimates of convergence for bv solutions it shows odd order finite volume advection schemes are better than even order finite volume schemes therefore the gibbs phenomena is controlled for odd order finite volume schemes numerical experiments sustain the theoretical analysis in particular the oscillations of the lax wendroff scheme for small courant numbers are correlated with its non stability in l 1 a scheme of order three is proved to be stable in l 1 and l

finite volume evolution galerkin methods for hyperbolic systems the subject of the paper is the derivation and analysis of new multidimensional high resolution finite volume evolution galerkin fveg schemes for systems of nonlinear hyperbolic conservation laws our approach couples a finite volume formulation with approximate evolution operators the latter are constructed using the bicharacteristics of the multidimensional hyperbolic system such that all of the infinitely many directions of wave propagation are taken into account in particular we propose a new fveg scheme which is designed in such a way that for a linear wave equation system the approximate evolution operator calculates any one dimensional planar wave exactly this operator improves the stability of the fveg scheme considerably leading to a stability limit closer to 1 using the results obtained for the wave equation system a new approximate evolution operator for the linearized euler equations is also derived the integrals over the cell interfaces also need to be approximated with care in this case our choice of simpson s rule is guided by stability analysis of model problems second order resolution is obtained by means of a piecewise bilinear recovery numerical experiments confirm the accuracy and multidimensional behavior of the new scheme

higher order time integration with a local lax wendroff procedure for a central scheme on an overlapping grid we have implemented a high order lax wendroff type time integration for a central scheme on an overlapping grid for conservation law problems using a local iterative approach presented by dumbser et al jcp 2008 12 we extend a local high order spatial reconstruction on each cell to a local higher order space time polynomial on the cell we rewrite the central scheme in a fully discrete form to avoid volume integration in the space time domain the fluxes at cell interfaces are calculated directly via integrating a higher order space time reconstruction of the flux we compare this approach with the corresponding multi stage runge kutta time integration rk numerical results show that the new time integration is more cost effective

chains demi matroids and profiles it is shown that each chain of linear codes has an associated demi matroid a combinatorial structure that extends the notion of a vector matroid of a linear code these demi matroids are proven to determine important properties of the chain and it is shown that linear code chain duality is represented by demi matroid duality in a natural and yet surprising way profiles are defined for arbitrary demi matroids and thus for linear code chains which generalizes previous results in the literature and provides simple and transparent proofs thereof finally applications are given to encryption of messages through wire tap channels of type ii

numerical modeling of transient two dimensional viscoelastic waves this paper deals with the numerical modeling of transient mechanical waves in linear viscoelastic solids dissipation mechanisms are described using the generalized zener model no time convolutions are required thanks to the introduction of memory variables that satisfy local in time differential equations by appropriately choosing the relaxation parameters it is possible to accurately describe a large range of materials such as solids with constant quality factors the evolution equations satisfied by the velocity the stress and the memory variables are written in the form of a first order system of pdes with a source term this system is solved by splitting it into two parts the propagative part is discretized explicitly using a fourth order ader scheme on a cartesian grid and the diffusive part is then solved exactly jump conditions along the interfaces are discretized by applying an immersed interface method numerical experiments of wave propagation in viscoelastic and fluid media show the efficiency of this numerical modeling for dealing with challenging problems such as multiple scattering configurations

an efficient hyperbolic relaxation system for dispersive non hydrostatic water waves and its solution with high order discontinuous galerkin schemes abstract in this paper we propose a novel set of first order hyperbolic equations that can model dispersive non hydrostatic free surface flows the governing pde system is obtained via a hyperbolic approximation of the family of non hydrostatic free surface flow models recently derived by sainte marie et al in 1 our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth averaged non hydrostatic pressure similar to the hyperbolic divergence cleaning applied in generalized lagrangian multiplier methods glm for magnetohydrodynamics mhd we suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing pde we then use arbitrary high order accurate ader discontinuous galerkin dg finite element schemes with an a posteriori subcell finite volume limiter in order to solve the proposed pde system numerically the final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet dry fronts it is well balanced making use of a path conservative formulation of hll type riemann solvers based on the straight line segment path furthermore the proposed ader dg scheme with a posteriori subcell finite volume limiter adapts very well to modern gpu architectures resulting in a very accurate robust and computationally efficient computational method for non hydrostatic free surface flows the new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions in all cases the achieved agreement with analytical solutions or experimental data is very good thus showing the validity of both the proposed mathematical model and the numerical solution algorithm

global existence of small solutions to a class of nonlinear evolution equations on etudie le comportement global des solutions d une classe d equations d evolution non lineaires on considere le probleme aux valeurs initiales pour des equations quasilineaires qui sont des perturbations d equations lineaires dissipatives

some estimates of solutions for the equations of motion of compressible viscous fluid in the three dimensional exterior domain abstract we consider the equations of motion of compressible viscous fluid in an exterior domain in r 3 we give the l q l p estimates for solutions to the linearized equations and show an optimal decay estimate for solutions to the nonlinear problem in particular we shall give l 1 estimates which implies the diffusion wave of compressible navier stokes flow

multi moment ader taylor methods for systems of conservation laws with source terms in one dimension a new integration method combining the ader time discretization with a multi moment finite volume framework is introduced ader runtime is reduced by performing only one cauchy kowalewski c k procedure per cell per time step and by using the differential transform method for high order derivatives three methods are implemented 1 single moment weno weno 2 two moment hermite weno hweno and 3 entirely local multi moment mm loc mm loc evolves all moments sharing the locality of galerkin methods yet with a constant time step during p refinement five 1 d experiments validate the methods 1 linear advection 2 burger s equation shock 3 transient shallow water sw 4 steady state sw simulation and 5 sw shock weno and hweno methods showed expected polynomial h refinement convergence and successfully limited oscillations for shock experiments mm loc showed expected polynomial h refinement and exponential p refinement convergence for linear advection and showed sub exponential yet super polynomial convergence with p refinement in the sw case hweno accuracy was generally equal to or better than a five moment mm loc scheme mm loc was less accurate than rkdg at lower refinements but with greater h and p convergence rkdg accuracy is eventually surpassed the ader time integrator of mm loc also proved more accurate with p refinement at a cfl of unity than a semi discrete rk analog of mm loc being faster in serial and requiring less frequent inter node communication than galerkin methods the ader based mm loc and hweno schemes can be spatially refined and have the same runtime making them a competitive option for further investigation

initial boundary value problems for the equations of motion of compressible viscous and heat conductive fluids the equations of motion of compressible viscous and heat conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region the global solution in time is proved to exist uniquely and approach the stationary state ast provided the prescribed initial data and the external force are sufficiently small

a third order accurate direct eulerian grp scheme for the euler equations in gas dynamics the paper proposes and implements a third order accurate direct eulerian generalized riemann problem grp scheme for one and two dimensional 1d 2d euler equations in gas dynamics it is an extension of the second order accurate grp scheme proposed in ben artzi et al 2006 5 the approximate states in numerical fluxes of the third order accurate grp scheme are derived by using the higher order weno reconstruction of the initial data the limiting values of the time derivatives of the solutions at the singularity point and the jacobian matrix besides the limiting values of the first order time derivatives of fluid variables the second order time derivatives are also needed in developing the present grp scheme and obtained by directly and analytically resolving the local grp in the eulerian formulation via two main ingredients i e the riemann invariants and rankine hugoniot jump conditions unfortunately for the sonic case that the transonic rarefaction wave appears in the grp the jacobian matrix is singular on the sonic line to this end those approximate states are given in a different way that is based on the analytical resolution of the transonic rarefaction wave and the local quadratic polynomial interpolation the 2d grp scheme is implemented by using the third order accurate time splitting method several numerical examples are given to demonstrate the accuracy and effectiveness of the proposed grp scheme in comparison to the second order accurate grp scheme

towards a more accurate flux corrected transport algorithm abstract an algorithm is discussed for solving first order hyperbolic equations whose solution may exhibit very steep and changing gradients shock like and large dynamic range in developing the algorithm emphasis has been placed on obtaining more accurate solutions the algorithm is based on flux corrected transport techniques

computation of reactive duct flows in external fields abstract a grp scheme is introduced for the numerical integration of the euler system of equations of compressible reactive flow in a duct of variable cross section subject to an external potential the grp generalized riemann problem scheme is based on an analytic solution of the grp at jump discontinuities it is a second order scheme generalizing the first order godunov scheme having the property of high resolution of shocks and other discontinuities some numerical examples are considered including an infinite spherical reflected shock a spherical blast wave and gas collapse under an external potential

latroids and their representation by codes over modules it has been known for some time that there is a connection between linear codes over fields and matroids represented over fields in fact a generator matrix for a linear code over a field is also a representation of a matroid over that field there are intimately related operations of deletion contraction minors and duality on both the code and the matroid the weight enumerator of the code is an evaluation of the tutte polynomial of the matroid and a standard identity relating the tutte polynomials of dual matroids gives rise to a macwilliams identity relating the weight enumerators of dual codes more recently codes over rings and modules have been considered and macwilliams type identities have been found in certain cases in this paper we consider codes over rings and modules with code duality based on a morita duality of categories of modules to these we associate latroids defined here we generalize notions of deletion contraction minors and duality on both codes and latroids and examine all natural relations among these we define generating functions associated with codes and latroids and prove identities relating them generalizing above mentioned generating functions and identities

ader schemes for the shallow water equations in channel with irregular bottom elevation this paper deals with the construction of high order ader numerical schemes for solving the one dimensional shallow water equations with variable bed elevation the non linear version of the schemes is based on eno reconstructions the governing equations are expressed in terms of total water height instead of total water depth and discharge the eno polynomial interpolation procedure is also applied to represent the variable bottom elevation ader schemes of up to fifth order of accuracy in space and time for the advection and source terms are implemented and systematically assessed with particular attention to their convergence rates non oscillatory results are obtained for discontinuous solutions both for the steady and unsteady cases the resulting schemes can be applied to solve realistic problems characterized by non uniform bottom geometries

numerical modeling of the acoustic wave propagation across a homogenized rigid microstructure in the time domain homogenization of a thin micro structure yields effective jump conditions that incorporate the geometrical features of the scatterers these jump conditions apply across a thin but nonzero thickness interface whose interior is disregarded this paper aims i to propose a numerical method able to handle the jump conditions in order to simulate the homogenized problem in the time domain ii to inspect the validity of the homogenized problem when compared to the real one for this purpose we adapt an immersed interface method originally developed for standard jump conditions across a zero thickness interface doing so allows us to handle arbitrary shaped interfaces on a cartesian grid with the same efficiency and accuracy of the numerical scheme than those obtained in an homogeneous medium numerical experiments are performed to test the properties of the numerical method and to inspect the validity of the homogenization problem

brain venous haemodynamics neurological diseases and mathematical modelling a review behind medicine m is physiology p behind physiology is physics p and behind physics is always mathematics m for which i expect that the symmetry of the quadruplet mppm will be compatible with the characteristic bias of hyperbolic partial differential equations a theme of this paper i start with a description of several idiopathic brain pathologies that appear to have a strong vascular dimension of which the most prominent example considered here is multiple sclerosis the most common neurodegenerative disabling disease in young adults other pathologies surveyed here include retinal abnormalities transient global amnesia transient monocular blindness meniere s disease and idiopathic parkinson s disease it is the hypothesised vascular aspect of these conditions that links medicine to mathematics through fluid mechanics in very complex networks of moving boundary blood vessels the second part of this paper is about mathematical modelling of the human cardiovascular system with particular reference to the venous system and the brain a review of a recently proposed multi scale mathematical model then follows consisting of a one dimensional hyperbolic description of blood flow in major arteries and veins coupled to a lumped parameter description of the remaining main components of the human circulation derivation and analysis of the hyperbolic equations is carried out for blood vessels admitting variable material properties and with emphasis on the venous system a much neglected aspect of cardiovascular mathematics veins unlike their arterial counterparts are highly deformable even collapsible under mild physiological conditions we address mathematical and numerical challenges regarding the numerical analysis of the hyperbolic pdes we deploy a modern non linear finite volume method of arbitrarily high order of accuracy in both space and time the ader methodology in vivo validation examples and brain haemodynamics computations are shown we also point out two preliminary but important new findings through the use of mathematical models namely that extracranial venous strictures produce chronic intracranial venous hypertension and that augmented pressure increases the blood vessel wall permeability

wave simulation in 2d heterogeneous transversely isotropic porous media with fractional attenuation a cartesian grid approach a time domain numerical modeling of transversely isotropic biot poroelastic waves is proposed in two dimensions the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce shifted fractional derivatives of order 1 2 involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations resulting in the biot da diffusive approximation model the properties of both the biot jkd and the biot da models are analyzed hyperbolicity decrease of energy dispersion to determine the coefficients of the diffusive approximation two approaches are analyzed gaussian quadratures and optimization methods in the frequency range of interest the nonlinear optimization is shown to be the better way of determination a splitting strategy is then applied to approximate numerically the biot da equations the propagative part is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly an immersed interface method is implemented to take into account heterogeneous media on a cartesian grid and to discretize the jump conditions at interfaces numerical experiments are presented comparisons with analytical solutions show the efficiency and the accuracy of the approach and some numerical experiments are performed to investigate wave phenomena in complex media such as multiple scattering across a set of random scatterers

arbitrary high order discontinuous galerkin schemes based on the grp method for compressible euler equations combined with the discontinuous galerkin dg framework the generalized riemann problem grp method is applied to design a grp dg scheme with high order accuracy for compressible euler equations since numerical fluxes with second order accuracy in time are derived by the grp method the reconstruction steps for physical variables in the new scheme are halved compared with the traditional runge kutta discontinuous galerkin rk dg scheme the numerical results are also improved due to more introduced physical information several numerical examples verify the validity of the proposed schemes

biot jkd model simulation of 1d transient poroelastic waves with fractional derivatives a time domain numerical modeling of biot poroelastic waves is presented the viscous dissipation occurring in the pores is described using the dynamic permeability model developed by johnson koplik dashen jkd some of the coefficients in the biot jkd model are proportional to the square root of the frequency in the time domain these coefficients introduce order 1 2 shifted fractional derivatives involving a convolution product based on a diffusive representation the convolution kernel is replaced by a finite number of memory variables that satisfy local in time ordinary differential equations thanks to the dispersion relation the coefficients in the diffusive representation are obtained by performing an optimization procedure in the frequency range of interest a splitting strategy is then applied numerically the propagative part of biot jkd equations is discretized using a fourth order ader scheme on a cartesian grid whereas the diffusive part is solved exactly comparisons with analytical solutions show the efficiency and the accuracy of this approach

solution of the riemann problem for irreversibly compressible media the riemann problem for an irreversibly compressible medium has been solved this solution introduces the maximum medium density that is attained in the process of active loading to close the system of equations an extra equation is required to describe the transfer of this parameter through the eulerian mesh the possible wave configurations have been analyzed and the corresponding equations for the evaluation of the contact pressure and velocity have been obtained the existence and uniqueness of the solution have been proven the technique of the riemann problem s solution for the arbitrary lagrange euler mesh was developed examples of the riemann problem solution for various wave configurations show that neglecting the bulk elastic plastic deformations yields significant errors in the results both quantitatively and qualitatively copyright 2009 john wiley sons ltd

modeling 1 d elastic p waves in a fractured rock with hyperbolic jump conditions the propagation of elastic waves in a fractured rock is investigated both theoretically and numerically outside the fractures the propagation of compressional waves is described in the simple framework of 1 d linear elastodynamics the focus here is on the interactions between the waves and fractures for this purpose the mechanical behavior of the fractures is modeled using nonlinear jump conditions deduced from the bandis barton model classically used in geomechanics well posedness of the initial boundary value problem thus obtained is proved numerical modeling is performed by coupling a time domain finite difference scheme with an interface method accounting for the jump conditions the numerical experiments show the effects of contact nonlinearities the harmonics generated may provide a nondestructive means of evaluating the mechanical properties of fractures

large time behavior of the semigroup on lp spaces associated with the linearized compressible navier stokes equation in a cylindrical domain abstract large time behavior of solutions to the linearized compressible navier stokes equation around the motionless state in a cylindrical domain is investigated the l p decay estimates of the associated semigroup are established for all 1 p it is also shown that the time asymptotic leading part of the semigroup is given by a one dimensional heat semigroup

an entropy satisfying muscl scheme for systems of conservation laws for the high order numerical approximation of hyperbolic systems of conservation laws we propose to use as a building principle an entropy diminishing criterion instead of the familiar total variation diminishing criterion introduced by harten for scalar equations based on this new criterion we derive entropy diminishing projections that ensure both the second order of accuracy and all of the classical discrete entropy inequalities the resulting scheme is a nonlinear version of the classical van leer s muscl scheme strong convergence of this second order entropy satisfying scheme is proved for systems of two equations numerical tests demonstrate the interest of our theory