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

hyperbolic conservation laws with relaxation the effect of relaxation is important in many physical situations it is present in the kinetic theory of gases elasticity with memory gas flow with thermo non equilibrium water waves etc the governing equations often take the form of hyperbolic conservation laws with lower order terms in this article we present and analyze a simple model of hyperbolic conservation laws with relaxation effects dynamic subcharacteristics governing the propagation of disturbances over strong wave forms are identified stability criteria for diffusion waves expansion waves and traveling waves are found and justified nonlinearly time asymptotic expansion and the energy method are used in the analysis for dissipative waves the expansion is similar in spirit to the chapman enskog expansion in the kinetic theory for shock waves however a different approach is needed

error bounds for the methods of glimm godunov and leveque the expected error in l 1 r at time t for glimm s scheme when applied to a scalar conservation law is bounded by left h frac 2 sqrt 3 left frac h delta t right 1 2 right left u 0 right bv bf r where h is the mesh size and delta t is the time step we show that the error in godunov s scheme is bounded by the same expression and use this result to analyze a variant of a large time step numerical method of leveque that may be viewed as a combination of godunov s scheme and a method of dafermos

probabilistic analysis of the upwind scheme for transport equations we provide a probabilistic analysis of the upwind scheme for d dimensional transport equations we associate a markov chain with the numerical scheme and then obtain a backward representation formula of kolmogorov type for the numerical solution we then understand that the error induced by the scheme is governed by the fluctuations of the markov chain around the characteristics of the flow we show in various situations that the fluctuations are of diffusive type as a by product we recover recent results due to merlet and vovelle numer math 106 129 155 2007 and merlet siam j numer anal 46 1 124 150 2007 we prove that the scheme is of order 1 2 in l infty 0 t l 1 mathbb r d for an integrable initial datum of bounded variation and of order 1 2 e for all e 0 in l infty 0 t times mathbb r d for an initial datum of lipschitz regularity our analysis provides a new interpretation of the numerical diffusion phenomenon

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

local error estimates for discontinuous solutions of nonlinear hyperbolic equations let u x t be the possibly discontinuous entropy solution of a nonlinear scalar conservation law with smooth initial data suppose u varepsilon x t is the solution of an approximate viscosity regularization where varepsilon 0 is the small viscosity amplitude it is shown that by post processing the small viscosity approximation u varepsilon pointwise values of u and its derivatives with an error as close to e as desired can be recovered the analysis relies on the adjoint problem of the forward error equation which in this case amounts to a backward linear transport equation with discontinuous coefficients the novelty of our approach is to use a generalized e condition of the forward problem in order to deduce a w 1 infty energy estimate for the discontinuous backward transport equation this in turn leads to e uniform estimate on moments of the error u varepsilon u the approach presented does not follow the characteristics and therefore applies mutatis mutandis to o

a robust numerical integrator for the short pulse equation near criticality abstract the short pulse equation was introduced by schafer wayne 2004 for modeling the propagation of ultrashort optical pulses while it can describe a wide range of solutions its ultrashort pulse solutions with a few cycles which the conventional nonlinear schrodinger equation does not possess have drawn much attention in such a region the solutions can become quite close to a singularity and thus existing numerical methods cease to work stably or even if they do they require high computational cost in this paper we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure preserving methods the resulting scheme successfully works even when the singularity occurs and thus gives a highly robust method for resolving near singular solutions of interest it is confirmed by numerical experiments and some new insights about derivative blow up are obtained

the convergence rate of approximate solutions for nonlinear scalar conservation laws let v varepsilon x t varepsilon 0 be a family of approximate solutions for the nonlinear scalar conservation law u t f u x 0 with c 0 1 initial data assume that v varepsilon x t are textit lip stable in the sense that they satisfy oleinik s e entropy condition it is shown that if these approximate solutions are textit lip consistent i e if v varepsilon cdot 0 u cdot 0 textit lip x v t varepsilon f v varepsilon x textit lip x t mathcal o varepsilon then they converge to the entropy solution and the convergence rate estimate v varepsilon cdot t u cdot t textit lip x mathcal o varepsilon holds consequently the familiar l p type and new pointwise error estimates are derived these convergence rate results are demonstrated in the context of entropy satisfying finite difference and glimm s schemes

design of pulse width modulation controllers for stabilization and tracking in derived dc to dc power converters feedback controllers based on pulse width modulation pwm are derived for the average input current stabilization and tracking problems in derived dc to dc power supplies of the buck and boost types the stabilization problems are solved on the basis of steady state considerations about the current ripple and exactly discretized nonlinear models describing the sampled pwm regulated input current trajectories in the boost derived converter the stabilization problem leads to an implicit static nonlinear feedback controller or duty ratio synthesizer which requires online solutions of a transcendental equation at each sampling instant the signal tracking problems are solved on the basis of discrete time non kalmanian state representation models describing the average pwm regulated input current these generalized state models naturally allow for explicit dynamical rather than static feedback regulators computer simulations including unmodelled load variations and external stochastic perturbation inputs are presented which test the robustness of the proposed pwm controller performances

exact boundary conditions for the initial value problem of convex conservation laws the initial value problem of convex conservation laws which includes the famous burgers inviscid equation plays an important rule not only in theoretical analysis for conservation laws but also in numerical computations for various numerical methods for example the initial value problem of the burgers equation is one of the most popular benchmarks in testing various numerical methods but in all the numerical tests the initial data have to be assumed that they are either periodic or having a compact support so that periodic boundary conditions at the periodic boundaries or two constant boundary conditions at two far apart spatial artificial boundaries can be used in practical computations in this paper for the initial value problem with any initial data we propose exact boundary conditions at two spatial artificial boundaries which contain a finite computational domain by using the lax s exact formulas for the convex conservation laws the well posedness of the initial boundary problem is discussed and the finite difference schemes applied to the artificial boundary problems are described numerical tests with the proposed artificial boundary conditions are carried out by using the lax friedrichs monotone difference schemes

differential algebraic approach in non linear dynamical compensator design for d c to d c power converters abstract a non linear dynamical compensator design based on fliess generalized controller canonical form for non linear dynamical systems is proposed for the asymptotic stabilization of output load voltage or current in pulse width modulation controlled d c to d c power supplies the technique rests on the recently introduced differential algebraic approach to the study of controlled dynamical systems

optimal l 1 rate of convergence for the viscosity method and monotone scheme to piecewise constant solutions with shocks we derive optimal error bounds for the viscosity method and monotone difference schemes to an initial value problem of scalar conservation laws with initial data being a finite number of piecewise constants subject to the initial discontinuities satisfying the entropy conditions it is known that the entropy solution of the problem is piecewise constant with a finite number of interacting shocks satisfying the entropy conditions a rigorous analysis shows that both the viscosity method and monotone schemes to approach the initial value problem have uniform l 1 error bounds of o epsilon and o delta x for the time infty t 0 respectively where epsilon and delta x are their corresponding viscosity coefficient and discrete mesh length the results are improvements over the half order rates of l 1 convergence numerical experiments for the lax friedrichs scheme are presented and numerical results justify the theoretical analysis

weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions abstract in this paper the weakly nonlinear limit for the relaxation approximation of conservation laws in several space dimensions is derived through asymptotic expansions and justified by employing the energy estimates compared with the work of g q chen c d levermore and t p liu 1994 comm pure appl math 47 787 830 the main difficulty we confront in our analyzes lies in the fact that the problem of the existence of an convex entropy which was essential in the proof of their estimate 5 29 for the system under our consideration is still an open problem since their theorem 3 2 was proved only for the 2 2 case

on the convergence of finite difference methods for pde under temporal refinement abstract in this note we present an issue in the verification of order of convergence for finite difference approximations to partial differential equations discretized in time and space when one refines both space and time together the convergence rate for a correct implementation will match that expected from an analysis of the asymptotic convergence rate or local truncation error of the method sometimes however only the time step size is reduced while the spatial mesh is held fixed the observed rate of convergence in this case may then differ from the formal order of accuracy of the method in space time in particular one class of methods second order lax wendroff time differencing produces only first order when the time step is refined at fixed spatial mesh resolution because the time and space differencing are intrinsically linked our method to arrive at this result is through analyzing the error at a defined point in time associated with many time steps we demonstrate our results computationally

convergence analysis of wavelet schemes for convection reaction equations under minimal regularity assumptions in this paper we analyze convergence rates of wavelet schemes for time dependent convection reaction equations within the framework of the eulerian lagrangian localized adjoint method ellam under certain minimal assumptions that guarantee h 1 regularity of exact solutions we show that a generic ellam scheme has a convergence rate mathcal o h sqrt delta t delta t in l 2 norm then applying the theory of operator interpolation we obtain error estimates for initial data with even lower regularity namely it is shown that the error of such a scheme is mathcal o h sqrt delta t theta delta t theta for initial data in a besov space displaystyle b theta 2 q 0 theta 1 0

probabilistic analysis of the upwind scheme for transport we provide a probabilistic analysis of the upwind scheme for multi dimensional transport equations we associate a markov chain with the numerical scheme and then obtain a backward representation formula of kolmogorov type for the numerical solution we then understand that the error induced by the scheme is governed by the fluctuations of the markov chain around the characteristics of the flow we show in various situations that the fluctuations are of diffusive type as a by product we prove that the scheme is of order 1 2 for an initial datum in bv and of order 1 2 a for all a 0 for a lipschitz continuous initial datum our analysis provides a new interpretation of the numerical diffusion phenomenon

error estimate for the finite volume scheme applied to the advection equation we study the convergence of a finite volume scheme for the linear advection equation with a lipschitz divergence free speed in r d we prove a h 1 2 error estimate in the l infty 0 t l 1 norm for bv data this result was expected from numerical experiments and is optimal

error estimate and the geometric corrector for the upwind finite volume method applied to the linear advection equation this paper deals with the upwind finite volume method applied to the linear advection equation on a bounded domain and with natural boundary conditions we introduce what we call the geometric corrector which is a sequence associated with every finite volume mesh in mathbf r nd and every nonvanishing vector mathbf a of mathbf r nd first we show that if the continuous solution is regular enough and if the norm of this corrector is bounded by the mesh size then an order one error estimate for the finite volume scheme occurs afterwards we prove that this norm is indeed bounded by the mesh size in several cases including the one where an arbitrary coarse conformal triangular mesh is uniformly refined in two dimensions computing numerically exactly this corrector allows us to state that this result might be extended under conditions to more general cases such as the one with independent refined meshes

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

a new glimm functional and convergence rate of glimm scheme for general systems of hyperbolic conservation laws in this paper we introduce a new glimm functional for general systems of hyperbolic conservation laws this new functional is consistent with the classical glimm functional for the case when each characteristic field is either genuinely nonlinear or linearly degenerate so that it can be viewed as optimal in some sense with this new functional the consistency of the glimm scheme is proved clearly for general systems moreover the convergence rate of the glimm scheme is shown to be the same as the one obtained in bressan marson arch ration mech anal 142 2 155 176 1998 for systems with each characteristic field being genuinely nonlinear or linearly degenerate

a hamiltonian preserving scheme for the liouville equation of geometrical optics with partial transmissions and reflections we construct a class of hamiltonian preserving numerical schemes for the liouville equation of geometrical optics with partial transmissions and reflections this equation arises in the high frequency limit of the linear wave equation with a discontinuous index of refraction in our previous work hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds j comput phys 214 2006 pp 672 697 we introduced the hamiltonian preserving schemes for the same equation when only complete transmissions or reflections occur at the interfaces these schemes are extended in this paper to the general case of partial transmissions and reflections the key idea is to build into the numerical flux the behavior of waves at the interface namely partial transmissions and reflections that satisfy snell s law of refraction with the correct transmission and reflection coefficients this scheme allows a hyperbolic stability condition under which positivity and stabilities in both l 1 and l infty norms are established numerical experiments are carried out to study the numerical accuracy

transient l1 error estimates for well balanced schemes on non resonant scalar balance laws abstract the ability of well balanced wb schemes to capture very accurately steady state regimes of non resonant hyperbolic systems of balance laws has been thoroughly illustrated since its introduction by greenberg and leroux 1996 15 see also the anterior wb glimm scheme in e 1992 8 this paper aims at showing by means of rigorous c t 0 l x 1 estimates that these schemes deliver an increased accuracy in transient regimes too namely after explaining that for the vast majority of non resonant scalar balance laws the c t 0 l x 1 error of conventional fractional step tang and teng 1995 45 numerical approximations grows exponentially in time like exp max g t x as a consequence of the use of gronwall s lemma it is shown that wb schemes involving an exact riemann solver suffer from a much smaller error amplification thanks to strict hyperbolicity their error grows at most only linearly in time see also layton 1984 30 numerical results on several test cases of increasing difficulty including the classical leveque yee s benchmark problem leveque and yee 1990 34 in the non stiff case confirm the analysis

a convergent scheme for a non local hamilton jacobi equation modelling dislocation dynamics we study dislocation dynamics with a level set point of view the model we present here looks at the zero level set of the solution of a non local hamilton jacobi equation as a dislocation in a plane of a crystal the front has a normal speed depending on the solution itself we prove existence and uniqueness for short time in the set of continuous viscosity solutions we also present a first order finite difference scheme for the corresponding level set formulation of the model the scheme is based on monotone numerical hamiltonian proposed by osher and sethian the non local character of the problem makes it not monotone we obtain an explicit convergence rate of the approximate solution to the viscosity solution we finally provide numerical simulations

the sharpness of kuznetsov s o d x l 1 error estimate for monotone difference schemes we derive a lower error bound for monotone difference schemes to the solution of the linear advection equation with bv initial data a rigorous analysis shows that for any monotone difference scheme the lower l 1 error bound is o x where x is the spatial stepsize

one sided stability and convergence of the nessyahu tadmor scheme non oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws the nessyahu tadmor nt scheme is an example of a second order scheme that is both robust and simple in this paper we prove a new stability property of the nt scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law this property is similar to the one sided lipschitz condition for first order schemes using this new stability we derive the convergence of the nt scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method we also derive an error estimate for monotone initial data

on exact solutions of the schafer wayne short pulse equation wki eigenvalue problem we consider a new equation recently found by sch afer and wayne hereafter named schshort pulse equation swspe describing the propagation of an ultrashort pulse in nonlinear media using some vanishing boundary conditions we construct and discuss the n soliton solutions to the previous equation by means of the wadati konno ichikawa wki method which is arguably more direct than the map through the sine gordon equation investigated much earlier by sakovich and sakovich 2005 j phys soc japan 74 239 2006 j phys a math gen 39 l361 we particularly focus our attention on the two soliton solution as an application as a result the collision process for two soliton solutions with similar amplitudes exhibits very different behaviour from the case when the amplitudes are dissimilar

a geometric approach to pulse width modulated control in nonlinear dynamical systems the author demonstrates under the assumption of high frequency control switchings the existence of an ideal equivalence among sliding regimes of variable structure feedback control and pulse width modulated pwm control responses in nonlinear dynamical systems this equivalence constitutes the basis for a geometric approach approach to pwm control loops design an illustrate example of energy conversion in a lossless switched controlled bilinear network is presented

high resolution nonoscillatory central schemes for hamilton jacobi equations in this paper we construct second order central schemes for multidimensional hamilton jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle thus these schemes provide the first examples of nonoscillatory second order godunov type schemes based on global projection operators numerical experiments are performed l 1 l infty errors and convergence rates are calculated for convex hamiltonians numerical evidence confirms that our central schemes converge with second order rates when measured in the l1 norm advocated in our recent paper numer math to appear the standard l infty norm however fails to detect this second order rate

error estimates for finite volume approximations of classical solutions for nonlinear systems of hyperbolic balance laws we consider a general class of finite volume schemes on unstructured but quasi uniform meshes for first order systems of hyperbolic balance laws on unstructured meshes provided the system is equipped with at least one entropy entropy flux tuple and the associated cauchy problem allows for a classical solution u we give conditions such that the finite volume approximation u h converges to u if the mesh parameter h tends to zero in fact we prove an error estimate of the form u u h l 2 le c sqrt h where c is independent of h the proof relies on a stability result for classical solutions in the class of entropy solutions due to dafermos arch rational mech anal 94 1979 pp 373 389 and diperna indiana univ math j 28 1979 pp 137 188 finally we present examples such that the conditions to apply the general convergence estimate can be satisfied at least in part the examples cover general scalar equations weakly coupled systems and the system of elastodynamics in one dimension moreover we generalize the concept of entropy conservative methods due to tadmor math comp 49 1987 pp 91 103 and show how this can be used to establish the convergence of finite volume methods for the system s case

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

viscous limits for piecewise smooth solutions to systems of conservation laws in this paper we study the zero dissipation problem for a general system of conservation laws with positive viscosity it is shown that if the solution of the problem with zero viscosity is piecewise smooth with a finite number of noninteracting shocks satisfying the entropy condition then there exist solutions to the corresponding system with viscosity that converge to the solutions of the system without viscosity away from shock discontinuities at a rate of order e as the viscosity coefficient e goes to zero the proof uses a matched asymptotic analysis and an energy estimate related to the stability theory for viscous shock profiles

existence of discrete shock profiles of a class of monotonicity preserving schemes for conservation laws when shock speed s times t x is rational the existence of solutions of shock profile equations on bounded intervals for monotonicity preserving schemes with continuous numerical flux is proved a sufficient condition under which the above solutions can be extended to j implying the existence of discrete shock profiles of numerical schemes is provided a class of monotonicity preserving schemes including all monotonicity preserving schemes with c 1 numerical flux functions the second order upwinding flux based muscl scheme the second order flux based muscl scheme with lax friedrichs splitting and the godunov scheme for scalar conservation laws are found to satisfy this condition thus the existence of discrete shock profiles for these schemes is established when s t x is rational

the short pulse equation and associated constraints the short pulse equation spe is considered as an initial boundary value problem it is found that the solutions of the spe must satisfy an integral relation otherwise the temporal derivative exhibits discontinuities this integral relation is not necessary for a solution to exist an infinite number of such constraints can be dynamically generated by the evolution equation

convergence rates for relaxation schemes approximating conservation laws in this paper we prove a global error estimate for a relaxation scheme approximating scalar conservation laws to this end we decompose the error into a relaxation error and a discretization error including an initial error omega ep we obtain the rate of convergence of sqrt ep in l1 for the relaxation step the estimate here is independent of the type of nonlinearity in the discretization step a convergence rate of sqrt del x in l1 is obtained these rates are independent of the choice of initial error omega ep thereby we obtain the order 1 2 for the total error

error estimate for finite volume scheme we study the convergence of a finite volume scheme for the linear advection equation with a lipschitz divergence free speed in r d we prove a h1 2 error estimate in the l 0 t l1 norm for bv data this result was expected from numerical experiments and is optimal

analysis of the implicit upwind finite volume scheme with rough coefficients we study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation in the low regularity diperna lions setting that is we are concerned with advecting velocity fields that are spatially sobolev regular and data that are merely integrable we prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at hbox least 1 2 the numerical error is estimated in terms of logarithmic kantorovich rubinstein distances and provides thus a bound on the rate of weak convergence

vanishing curvature viscosity for front propagation in this paper we study the front propagation with constant speed and small curvature viscosity we first investigate two related problems of conservation laws one of which is on the nonlinear viscosity methods for the conservation laws and the other one is on the structure of solutions to conservation laws with l 1 initial data we show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first order rate of l 1 convergence the solutions of conservation laws with l 1 initial data are shown to be bounded after t 0 if all singular points of initial data are from shocks these results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed as the small parameter goes to zero after the front breaks down the cusps will disappear promptly and corners will be formed 2000 academic press

on convergence of minmod type schemes a class of nonoscillatory numerical methods for solving nonlinear scalar conservation laws in one space dimension is considered this class of methods contains the classical lax friedrichs and the second order nessyahu tadmor schemes in the case of linear flux new l2 stability results and error estimates for the methods are proved numerical experiments confirm that these methods are one sided l2 stable for convex flux instead of the usual lip stability

reversible linear and nonlinear discrete time dynamics it is shown that any causal discrete time system can be realized with a reversible or invertible state variable representation this bridges the gap with continuous time where flows always satisfy this property and settles an old debate on the relevance of invertibility in nonlinear discrete dynamics as the result applies to the linear case it is shown how the classic approach should be slightly modified and several examples of the benefits that can be obtained are given difference algebra is employed for nonlinear systems it leads to a most elegant characterization of causality which enables reversibility to be demonstrated module theory suffices in the linear case which is supplemented by elementary matrix manipulations

a priori error estimates for approximate solutions to convex conservation laws summary we introduce a new technique for proving a priori error estimates between the entropy weak solution of a scalar conservation law and a finite difference approximation calculated with the scheme of engquistosher lax friedrichs or godunov this technique is a discrete counterpart of the duality technique introduced by tadmor siam j numer anal 1991 the error is related to the consistency error of cell averages of the entropy weak solution this consistency error can be estimated by exploiting a regularity structure of the entropy weak solution one ends up with optimal error estimates

verification of fluid dynamic codes in the presence of shocks and other discontinuities the verification that computer codes correctly solve their model equations is critical to the continued success of numerical simulation the method of manufactured solutions mms is the best method currently available for this kind of verification for differential equations however it cannot be used directly with discontinuous solutions as is required for the verification of high speed aerodynamic codes with shocks an integrative approach can extend the applicability of mms to both discontinuous solutions such as shocks or material interfaces as well as integral equations we present an implementation of integrative mms based on intelligent subdivision of integration domains that is both highly accurate and fast and results in a rigorous one step verification procedure for shock capturing codes numerical integration is found to be accurate to machine precision when tested on exact solutions of the linear heat equation and the euler equations even in the presence of discontinuous flow features intelligent subdivision of integration domains also improves computational performance by approximately 60i compared to the same algorithm without intelligent subdivisions we demonstrate the use of mms in the verification of the bacl streamer inviscid gas dynamics code integral mms is found to compute convergence rates that are equivalent to those computed using differential mms and comparable to those computed using discontinuous exact solutions suggesting integral mms is a valid method for verification of both integral and shock capturing codes

hamiltonian preserving schemes for the liouville equation of geometrical optics with discontinuous local wave speeds in this paper we construct two classes of hamiltonian preserving numerical schemes for a liouville equation with discontinuous local wave speed this equation arises in the phase space description of geometrical optics and has been the foundation of the recently developed level set methods for multivalued solution in geometrical optics we extend our previous work in s jin x wen hamiltonian preserving schemes for the liouville equation with discontinuous potentials commun math sci 3 2005 285 315 for the semiclassical limit of the schrodinger equation into this system the designing principle of the hamiltonian preservation by building in the particle behavior at the interface into the numerical flux is used here and as a consequence we obtain two classes of schemes that allow a hyperbolic stability condition when a plane wave hits a flat interface the hamiltonian preservation is shown to be equivalent to snell s law of refraction in the case when the ratio of wave length over the width of the interface goes to zero when both length scales go to zero positivity and stabilities in both l 1 and l norms are established for both schemes the approach also provides a selection criterion for a unique solution of the underlying linear hyperbolic equation with singular discontinuous and measure valued coefficients benchmark numerical examples are given with analytic solution constructed to study the numerical accuracy of these schemes

pointwise error estimates for scalar conservation laws with piecewise smooth solutions we introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation and in fact more general approximations to scalar conservation laws with piecewise smooth solutions to this end we derive a transport inequality for an appropriately weighted error function the key ingredient in our approach is a one sided interpolation inequality between classical l1 error estimates and lip stability bounds the one sided interpolation interesting for its own sake enables us to convert a global l1 result into a nonoptimal local estimate this in turn provides the necessary bounds on the coefficients of the above mentioned transport inequality estimates on the weighted error then follow from the maximum principle and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation unlike previous works in this direction our method can deal with finitely many waves with possible collisions moreover in our approach one does not follow the characteristics but instead makes use of the energy method and hence this approach could be extended to other types of approximate solutions

l 1 stability and error estimates for approximate hamilton jacobi solutions we study the l 1 stability and error estimates of general approximate solutions for the cauchy problem associated with multidimensional hamilton jacobi h j equations for strictly convex hamiltonians we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors we then demonstrate this general theory for two types of approximations approximate solutions constructed by the vanishing viscosity method and by godunov type finite difference methods if we let epsilon denote the small scale of such approximations the viscosity amplitude epsilon the spatial grad size delta x etc then our l 1 error estimates are of cal o epsilon and are sharper than the classical l infty results of order one half cal o sqrt epsilon the main building blocks of our theory are the notions of the semi concave stability condition and l 1 measure of the truncation error the whole theory could be viewed as a multidimensional extension of the lip prime stability theory for one dimensional nonlinear conservation laws developed by tadmor et al 34 24 25 in addition we construct new godunov type schemes for h j equations which consist of an exact evolution operator and a global projection operator here we restrict our attention to linear projection operators first order schemes we note however that our convergence theory applies equally well to nonlinear projections used in the context of modern high resolution conservation laws we prove semi concave stability and obtain l 1 bounds on their associated truncation errors l 1 convergence of order one then follows second order central godunov type schemes are also constructed numerical experiments are performed errors and orders are calculated to confirm our l 1 theory

approximation of mono dimensional hyperbolic systems a lattice boltzmann scheme as a relaxation method we focus on mono dimensional hyperbolic systems approximated by a particular lattice boltzmann scheme the scheme is described in the framework of the multiple relaxation times method and stability conditions are given an analysis is done to link the scheme with an explicit finite differences approximation of the relaxation method proposed by jin and xin several numerical illustrations are given for the transport equation burger s equation the p system and full compressible euler s system

on the regularity of approximate solutions to conservation laws with piecewise smooth solutions in this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted w1 1 space convergence rate for the derivative of the approximate solutions is established under the assumption that a weak pointwise error estimate is given in other words we are able to convert weak pointwise error estimates to optimal error bounds in a weighted w1 1 space for convex conservation laws the assumption of a weak pointwise error estimate is verified by tadmor siam j numer anal 28 1991 pp 891 906 therefore one immediate application of our w1 1 convergence theory is that for convex conservation laws we indeed have w1 1 error bounds for the approximate solutions to conservation laws furthermore the co en pointwise error estimates of tadmor and tang siam j numer anal 36 1999 pp 1739 1758 are recovered by the use of the w1 1 convergence result

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

l p l q decay estimates for dissipative linear hyperbolic systems in 1d abstract given a b r n n we consider the cauchy problem for partially dissipative hyperbolic systems having the form t u a x u b u 0 with the aim of providing a detailed description of the large time behavior sharp l p l q estimates for 1 q p are established for the distance between the solution to the system and a time asymptotic profile where the profile is the superposition of diffusion waves and exponentially decaying waves they show that the solution to the system decays to the diffusion waves 1 2 faster than the diffusive decay rate 1 q 1 p 2 while the exponential decaying waves are negligible for large time in particular under a symmetry property it decays 1 faster than the proof is based on a complex interpolation argument once l r estimates for the fundamental solution to the system in the frequency space and fourier multiplier estimates are accomplished

on the convergence rate of vanishing viscosity approximations given a strictly hyperbolic genuinely nonlinear system of conservation laws we prove the a priori boundu t u e t l1 o 1 1 t e ln e on the distance between an exact bv solution u and a viscous approximation u e letting the viscosity coefficient e 0 in the proof starting from u we construct an approximation of the viscous solution u e by taking a mollification u e and inserting viscous shock profiles at the locations of finitely many large shocks for each fixed e error estimates are then obtained by introducing new lyapunov functionals that control interactions of shock waves in the same family and also interactions of waves in different families c 2004 wiley periodicals inc

on the l2 and l1 convergence rates of viscous solutions of the keyfitz kranzer system with piecewise smooth and large bv data abstract we study the zero dissipation problem of the keyfitz kranzer system in l 2 and l 1 spaces when the solution of the inviscid problem is piecewise smooth and has finitely many noninteracting shocks with finite strength there exists for each e the viscosity unique solution to the viscous problem with modified initial data and it converges to the given inviscid solution away from shock discontinuities as e tends to zero convergence rates are given in terms of e the proof is given by a matched asymptotic analysis and a weighted elementary energy method

error estimates for the staggered lax friedrichs scheme on unstructured grids staggered grid finite volume methods also called central schemes were introduced in one dimension by nessyahu and tadmor in 1990 in order to avoid the necessity of having information on solutions of riemann problems for the evaluation of numerical fluxes we consider the general case in multidimensions and on general staggered grids which have to satisfy only an overlap assumption we interpret the staggered lax friedrichs scheme as a three step method consisting of a prolongation step onto a finer intersection grid a finite volume step with an arbitrarily good numerical flux e g godunov flux on the intersection grid followed by an averaging step such that the calculation of numerical fluxes reduces to evaluations of the continuous flux using this point of view we prove an a posteriori error estimate and an a priori error estimate in the l1 norm in space and time which is of order h1 4 where h is a mesh size parameter hence we recover for the staggered lax friedrichs scheme the same order of convergence as for upwind finite volume methods on a fixed grid

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

on convergence of a domain decomposition method for a scalar conservation law in this paper we prove convergence of a domain decomposition method for one dimensional scalar conservation laws by dealing carefully with nonconservative terms at the interface of subdomains the method consists of an explicit scheme in some subdomains and an implicit scheme in other subdomains with a numerical flux being the same as the one used in the explicit scheme although such a multidomain algorithm is not strictly conservative the conservation error ce 0 n delta t is equal to mathcal o delta t regardless of the smoothness of the solution finally two test examples are given to validate convergence and the computational efficiency of the present method

rarefactions and large time behavior for parabolic equations and monotone schemes we consider the large time behavior of monotone semigroups associated with degenerate parabolic equations and monotone difference schemes for an appropriate class of initial data the solution is shown to converge to rarefaction waves at a determined asymptotic rate

the optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws we are interested in the rate of convergence in l1 of the approximate solution of a conservation law generated by a monotone finite difference scheme kuznetsov has proved that this rate is 1 2 ussr comput math math phys 16 1976 pp 105 119 and topics numer anal iii in proc roy irish acad conf dublin 1976 pp 183 197 and recently teng and zhang have proved this estimate to be sharp for a linear flux siam j numer anal 34 1997 pp 959 978 we prove by constructing appropriate initial data for the cauchy problem that kuznetsov s estimates are sharp for a nonlinear flux as well

stochastic and variational approach to the lax friedrichs scheme we present a stochastic and variational aspect of the lax friedrichs scheme applied to hyperbolic scalar conservation laws this is a finite difference version of fleming s results 69 that the vanishing viscosity method is characterized by stochastic processes and calculus of variations we convert the difference equation into that of the hamilton jacobi type and introduce corresponding calculus of variations with random walks the stability of the scheme is obtained through the calculus of variations the convergence of approximation is derived from the law of large numbers in hyperbolic scaling limit of random walks the main advantages due to our approach are the following our framework is basically pointwise convergence not l1 as usual which yields uniform convergence except small neighborhoods of shocks the convergence proof is verified for arbitrarily large time interval which is hard to obtain in the case of flux functions of general types depending on both space and time the approximation of characteristics curves is available as well as that of pde solutions which is particularly important for applications of the lax friedrichs scheme to the weak kam theory

solitary wave solutions of the short pulse equation an exact nonsingular solitary wave solution of the schafer wayne short pulse equation is derived from the breather solution of the sine gordon equation by means of a transformation between these two integrable equations

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

on the piecewise smooth solutions to non homogeneous scalar conservation laws abstract we study the structure and smoothness of non homogeneous convex conservation laws the question regarding the number of smoothness pieces is addressed it is shown that under certain conditions on the initial data the entropy solution has only a finite number of discontinuous curves we also obtain some global estimates on derivatives of the piecewise smooth entropy solution along the generalized characteristics these estimates play important roles in obtaining the optimal rate of convergence for various approximation methods to conservation laws

modified equation for adaptive monotone difference schemes and its convergent analysis a modified parabolic equation for adaptive monotone difference schemes based on equal arclength mesh applied to the linear convection equation is derived and its convergence analysis shows that solutions of the modified equation approach a discontinuous piecewise smooth solution of the linear convection equation at order one rate in the l 1 norm it is well known that solutions of the monotone schemes with uniform meshes and their modified equation approach the same discontinuous solution at a half order rate in the l 1 norm therefore the convergence analysis for the modified equation provided in this work demonstrates theoretically that the monotone schemes with adaptive grids can improve the solution accuracy numerical experiments also confirm the theoretical conclusions

the convergence rate of godunov type schemes godunov type schemes form a special class of transport projection methods for the approximate solution of nonlinear hyperbolic conservation laws the authors study the convergence rate of such sche