existence and stability of stationary profiles of the lw scheme in this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation laws when a difference scheme introduces artificial numerical diffusion for example the lax friedrichs scheme we experience smearing of the shocks whereas when a scheme introduces numerical dispersion for example the lax wendroff scheme we experience oscillations which decay exponentially fast on both sides of the shock in his dissertation gray jennings studied approximation by monotone schemes these contain artificial viscosity and are first order accurate they are known to be contractive in the sense of any lp norm jennings showed existence and l1 stability of traveling discrete smeared shocks for such schemes here we study similar questions for the lax wendroff scheme without artificial viscosity this is a nonmonotone second order accurate scheme we prove existence of a one parameter family of stationary profiles we also prove stability of these profiles for small perturbations in the sense of a suitably weighted l2 norm the proof relies on studying the linearized lax wendroff scheme

convergence analysis for operator splitting methods applied to conservation laws with stiff source terms we analyze the order of convergence for operator splitting methods applied to conservation laws with stiff source terms we suppose that the source term q u is dissipative it is proved that the l 1 error introduced by the time splitting can be bounded by o d t vert q u 0 vert l 1 which is an improvement of the o q d t upper bound where d t is the splitting time step q is the lipschitz constant of q or q max u vert q u vert in case q is smooth a generic model with a special form of stiff source is also investigated we propose a nonuniform temporal mesh to eliminate the effect of the initial layer introduced by the stiff source term our results are derived by using parabolic regularizations rather than using kuznetsov s approximation theory which has been employed as a standard approach for error analysis to the dimensional or time splitting methods numerical examples are presented to illustrate the theoretical results

on a time dependent grade two fluid model in two dimensions we prove the existence of a weak solution of a time dependent grade two fluid model in a plane lipschitz domain and uniqueness of the solution in a convex polygon the method of proof is constructive and can be adapted to the numerical analysis of finite element schemes for solving this problem numerically

decay rate for perturbations of stationary discrete shocks for convex scalar conservation laws this paper is to study the decay rate for perturbations of stationary discrete shocks for the lax friedrichs scheme approximating the scalar conservation laws by means of an elementary weighted energy method if the summation of the initial perturbation over j is small and decays at the algebraic rate as j then the solution approaches the stationary discrete shock profiles at the corresponding rate as n this rate seems to be almost optimal compared with the rate in the continuous counterpart proofs are given by applying the weighted energy integration method to the integrated scheme of the original one the selection of the time dependent discrete weight function plays a crucial role in this procedure

the l 1 global decay to discrete shocks for scalar monotone schemes given a family of discrete shocks of a monotone scheme we prove that the discrete shock profile with rational shock speed is asymptotically stable with respect to the l1 topology 1 if u0 l1 then un n 1 0 as n under no restriction conditions of the initial data to the interval inf sup the asymptotic wave profile is uniquely identified from the above family by a mass parameter

an analysis of the rayleigh stokes problem for a generalized second grade fluid we study the rayleigh stokes problem for a generalized second grade fluid which involves a riemann liouville fractional derivative in time and present an analysis of the problem in the continuous space semidiscrete and fully discrete formulations we establish the sobolev regularity of the homogeneous problem for both smooth and nonsmooth initial data v v including v in l 2 omega v l2 a space semidiscrete galerkin scheme using continuous piecewise linear finite elements is developed and optimal with respect to initial data regularity error estimates for the finite element approximations are derived further two fully discrete schemes based on the backward euler method and second order backward difference method and the related convolution quadrature are developed and optimal error estimates are derived for the fully discrete approximations for both smooth and nonsmooth initial data numerical results for one and two dimensional examples with smooth and nonsmooth initial data are presented to illustrate the efficiency of the method and to verify the convergence theory

on the stability of finite element methods for shock waves this paper we study the large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws of the form 1 1 u t f u x u xx x 2 r i t 0 u 2 r i u delta 0 u 0 delta we treat systems which are strictly hyperbolic such systems possess a smooth travelling wave solution which is called a viscous p shock wave solution u x t oe x gamma oet x sigma1 oe x u sigma provided that the shock strength ffl j ju gamma u gamma j is small 19 the constant states u sigma and the wave speed oe are related by the rankine hugoniot condition 1 3a f u gamma gamma f u oe u gamma gamma u

error bound between monotone difference schemes and their modified equations it is widely believed that if monotone difference schemes are applied to the linear convection equation with discontinuous initial data then solutions of the monotone schemes are closer to solutions of their parabolic modified equations than that of the original convection equation we will confirm the conjecture in this paper it is well known that solutions of the monotone schemes and their parabolic modified equations approach discontinuous solutions of the linear convection equation at a rate only half in the l 1 norm we will prove that the error bound between solutions of the monotone schemes and that of their modified equations is order one in the l 1 norm therefore the conclusion shows that the monotone schemes solve the modified equations more accurately than the original convection equation even if the initial data is discontinuous as a consequence of the main result we will show that the half order rate of convergence for the monotone schemes to the convection equation is the best possible

nonlinear stability of discrete shocks for systems of conservation laws in this paper we study the asymptotic nonlinear stability of discrete shocks for the lax friedrichs scheme for approximating general m systems of nonlinear hyperbolic conservation laws it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in the lp norm for all p 1 provided that the sums of the initial perturbations equal zero these results should shed light on the convergence of the numerical solution constructed by the lax friedrichs scheme for the single shock solution of system of hyperbolic conservation laws if the riemann solution corresponding to the given far field states is a superposition of m single shocks from each characteristic family we show that the corresponding multiple discrete shocks are nonlinearly stable in l p p 2 these results are proved by using both a weighted estimate and a characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the lax friedrichs scheme

an investigation of the internal structure of shock profiles for shock capturing schemes the theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions so called continuum shock profiles however the previous papers have focused on linear methods we have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes including riemann solver based flux limiter methods high resolution central schemes and eno type methods in some cases no continuum profiles exists however in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle the possibility of such behavior was anticipated in bultelle grassin and serre 1998 but no specific examples or other evidence of this behavior have previously been given in other cases our results indicate that continuum shock profiles exist but are very complicated we also study phase space orbits with regard to post shock oscillations

asymptotic stability of traveling wave solutions of systems for one dimensional gas motion the asymptotic stability of traveling wave solutions with shock profile is investigated for several systems in gas dynamics 1 the solution of a scalar conservation law with viscosity approaches the traveling wave solution at the ratet for some 0 ast provided that the initial disturbance is small and of integral zero and in addition decays at an algebraic rate for x 2 the traveling wave solution with nishida and smoller s condition of the system of a viscous heat conductive ideal gas is asymptotically stable provided the initial disturbance is small and of integral zero 3 the traveling wave solution with weak shock profile of the broadwell model system of the boltzmann equation is asymptotically stable provided the initial disturbance is small and its hydrodynamical moments are of integral zero each proof is given by applying an elementary energy method to the integrated system of the conservation form of the original one the property of integral zero of the initial disturbance plays a crucial role in this procedure

stability analysis of discontinuous galerkin discrete shock profiles for steady scalar conservation laws we present a stability analysis of stationary discrete shock profiles for a discontinuous galerkin method approximating scalar nonlinear hyperbolic conservation laws with a convex flux we consider the godunov method to evaluate the numerical flux and use either an explicit third order runge kutta scheme or an implicit backward euler scheme for the time integration applying a linear stability analysis we show that the steady solutions may become unstable when the numerical flux is not differentiable in particular the situation of a shock at an interface of the mesh corresponds to an unstable solution for a space discretization accuracy higher than second order whatever the time integration method and the physical flux spectra of the linearized operator indicate that the fourth order numerical scheme with the explicit time integration is also unstable for a continuous range of shock positions around an interface

a lax shock profile satisfying a sufficient condition of spectral instability abstract the aim of this paper is to construct a spectrally unstable lax shock profile of the semi linear approximation of a 2 2 system of conservation laws and to show numerically the instability thanks to an appropriate scheme

one sided difference approximations for nonlinear conservation laws we analyze one sided or upwind finite difference approximations to hyperbolic partial differential equations and in particular nonlinear conservation laws second order schemes are designed for which we prove both nonlinear stability and that the entropy condition is satisfied for limit solutions we show that no such stable approximation of order higher than two is possible these one sided schemes have desirable properties for shock calculations we show that the proper switch used to change the direction in the upwind differencing across a shock is of great importance new and simple schemes are developed for which we prove qualitative properties such as sharp monotone shock profiles existence uniqueness and stability of discrete shocks numerical examples are given

stability of one dimensional boundary layers by using green s functions the aim of this paper is to investigate the stability of one dimensional boundary layers of parabolic systems as the viscosity goes to 0 in the noncharacteristic case and more precisely to prove that spectral stability implies linear and nonlinear stability of approximate solutions in particular we replace the smallness condition obtained by the energy method 10 13 by a weaker spectral condition 2001 john wiley sons inc

unstable godunov discrete profiles for steady shock waves given a system of n conservation laws ut f u x 0 the steady shock waves when processed by the godunov scheme admit rather simple discrete profiles one shows that the linear stability of these profiles depends only on the location of the eigenvalues of some endomorphism of an n 1 dimensional space applying our theory to the gas dynamics with the perfect gas law p ga 1 rho e we construct unstable profiles for values of ga between 1 and ga and rather strong shocks since ga in 7 5 5 3 our result applies to the air but not to monoatomic gases finally we illustrate this analysis by some numerical experiments both with ga ga

l 1 stability of stationary discrete shocks the nonlinear stability in the l sup p norm p ge 1 of stationary weak discrete shocks for the lax friedrichs scheme approximating general m times m systems of nonlinear hyperbolic conservation laws is proved provided that the summations of the initial perturbations equal zero the result is proved by using a both weighted estimate and characteristic energy method based on the internal structures of the discrete shocks and the essential monotonicity of the lax friedrichs scheme 13 refs

boundary layer stability in real vanishing viscosity limit in the previous paper 20 an evans function machinery for the study of boundary layer stability was developed there the analysis was restricted to strongly parabolic perturbations that is to an approximation of the form ut f u x b u ux x 1 with an elliptic matrix b however real models like the navier stokes approximation of the euler equations for a gas flow involve incompletely parabolic perturbations b is not invertible in general

convergence rates to the discrete travelling wave for relaxation schemes this paper is concerned with the asymptotic convergence of nu merical solutions toward discrete travelling waves for a class of relaxation nu merical schemes approximating the scalar conservation law it is shown that if the initial perturbations possess some algebraic decay in space then the nu merical solutions converge to the discrete travelling wave at a corresponding algebraic rate in time provided the sums of the initial perturbations for the u component equal zero a polynomially weighted l 2 norm on the perturba tion of the discrete travelling wave and a technical energy method are applied to obtain the asymptotic convergence rate

nonlinear stability of stationary discrete shocks for nonconvex scalar conservation laws this paper is to study the asymptotic stability of stationary discrete shocks for the lax friedrichs scheme approximating nonconvex scalar conservation laws provided that the summations of the initial perturbations equal to zero the result is proved by using a weighted energy method based on the nonconvexity moreover the l 1 stability is also obtained the key points of our proofs are to choose a suitable weight function

grade two fluids on non smooth domain driven by multiplicative noise existence uniqueness and regularity the author presented a lecture on the subject of this paper at the workshop on numerics for stochastic partial differential equations and their applications held at the johann radon institute for computational and applied mathematics ricam linz austria during the period 04 16 december 2016

convergence rates to discrete shocks for nonconvex conservation laws this paper is concerned with polynomial decay rates of perturbations to stationary discrete shocks for the lax friedrichs scheme approximating non convex scalar conservation laws we assume that the discrete initial data tend to constant states as j rightarrow pm infty respectively and that the riemann problem for the corresponding hyperbolic equation admits a stationary shock wave if the summation of the initial perturbation over infty j is small and decays with an algebraic rate as j rightarrow infty then the perturbations to discrete shocks are shown to decay with the corresponding rate as n rightarrow infty the proof is given by applying weighted energy estimates a discrete weight function which depends on the space time variables for the decay rate and the state of the discrete shocks in order to treat the non convexity plays a crucial role

existence and asymptotic stability of relaxation discrete shock profiles kin this paper we study the asymptotic nonlinear stability of discrete shocks of the relaxing scheme for approximating the general system of nonlinear hyperbolic conservation laws the existence of discrete shocks is established by suitable manifold construction and it is shown that weak single discrete shocks for such a scheme are nonlinearly stable in l 2 provided that the sums of the initial perturbations equal zero these results should shed light on the convergence of the numerical solution constructed by the relaxing scheme for the single shock solution of the system of hyperbolic conservation laws these results are proved by using both a weighted norm estimate and a characteristic energy method based on the internal structures of the discrete shocks

viscosity methods for piecewise smooth solutions to scalar conservation laws it is proved that for scalar conservation laws if the flux function is strictly convex and if the entropy solution is piecewise smooth with finitely many discontinuities which includes initial central rarefaction waves initial shocks possible spontaneous formation of shocks in a future time and interactions of all these patterns then the error of viscosity solution to the inviscid solution is bounded by o log in the l 1 norm which is an improvement of the o upper bound if neither central rarefaction waves nor spontaneous shocks occur the error bound is improved to o

existence and uniqueness of traveling waves and error estimates for godunov schemes of conservation laws the existence and uniqueness of the lipschitz continuous traveling wave of godunov s scheme for scalar conservation laws are proved the structure of the traveling waves is studied the approximation error of godunov s scheme on single shock solutions is shown to be o 1 x

lyapunov functions on product spaces and stability theory of delay differential equations if we examine the lyapunov functionals constructed for all the examples that have been discussed so far in the literature we find that the investigators inadvertently employ a combination of a lyapunov function and a functional in such a way that the corresponding derivative can be estimated suitably without demanding minimal classes of functions or the knowledge of solutions as in the case of lyapunov functions or lyapunov functionals respectively 3 61 the method of lyapunov functionals however demands the knowledge of solutions of delay differential equations considered and consequently the discussion of examples is not really in the spirit of the method this observation leads us to develop the method of lyapunov functions on product spaces for studying stability properties of equations with delay where except conceptually the knowledge of solutions is not demanded we shall also develop stability theory in terms of two measures which unifies several known stability concepts our presentation demonstrates the advantage of utilizing lyapunov functions on product spaces 391 0022 247x 91 3 00

upwind discretization of a time dependent two dimensional grade two fluid model in this paper we propose a finite element scheme for solving numerically the equations of a transient two dimensional grade two non newtonian rivlin ericksen fluid model this system of equations is considered an appropriate model for the motion of a water solution of polymers by introducing a new variable denoted z we split the problem into a coupled one with a transport equation as one of our aims is to derive unconditional a priori estimates from the discrete analogue of the transport equation we stabilize our scheme by adding a consistent stabilizing term we use the p 2 p 1 taylor hood finite element scheme for the velocity v and the pressure p and the discontinuous p 1 finite element for an auxiliary variable z the error is of the order of h 3 2 k considering that the discretization of the transport equation loses inevitably a factor h 1 2