We obtain sharp pointwise error estimates for relaxation approximation to scalar conservation laws with piecewise smooth solutions. We first prove that the first order partial derivatives for the perturbation solutions are uniformly upper bounded (the so-called Lip stability). An one-sided interpolation inequality between classical L bounds enables us to convert a global L result into a (non-optimal) local estimate. Optimal error bounds on the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems. The main difficulties in obtaining the Lip stability and the optimal pointwise errors are how to construct appropriate "difference functions" so that the maximum principle can be applied. Contents 1

Abstract. 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 L 1 error estimates and Lip + stability bounds. The one-sided interpolation, interesting for its own sake, enables us to convert a global L 1 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.

.<F3.818e+05> We analyze the order of convergence for operator splitting methods applied to conservation laws with sti# source terms. We suppose that the source term<F3.69e+05><F3.818e+05><F3.69e+05><F3.818e+05> q(u) is dissipative. It is proved that the<F3.69e+05> L<F2.742e+05> 1<F3.818e+05> error introduced by the time splitting can be bounded by<F3.69e+05><F3.818e+05><F3.69e+05><F3.714e+05><F3.69e+05><F3.818e+05><F3.69e+05> O(#t#q(u<F2.742e+05> 0<F3.818e+05><F3.714e+05> )#<F3.264e+05> L<F2.949e+05> 1<F3.818e+05> ), which is an improvement of the<F3.69e+05><F3.818e+05><F3.69e+05><F3.818e+05><F3.69e+05><F3.818e+05> O(Q#t) upper bound, where<F3.69e+05> #t<F3.818e+05> is the splitting time step,<F3.69e+05> Q<F3.818e+05> is the Lipschitz constant of<F3.69e+05><F3.818e+05> q, or<F3.69e+05> Q<F3.818e+05> =<F3.264e+05> maxu<F3.714e+05><F3.69e+05> |q<F3.659e+05> #<F3.818e+05><F3.69e+05><F3.818e+05><F3.714e+05> (u)|<F3.818e+05> in case<F3.69e+05> q<F3.818e+05> is smooth. A generic model w...

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(\Deltatkq(u 0)k L 1), which is an improvement of the O(Q\Deltat) upper bound, where \Deltat is the splitting time step, Q is the Lipschitz constant of q or Q = max u jq 0 (u)j in case q is smooth. A generic model with a special form of stiff source is also investigated. We propose a non-uniform 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 the 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.

. We introduce a new method to obtain pointwise error estimates for vanishing viscosity and finite difference approximations of scalar conservation laws with piecewise smooth solutions. This method can deal with finitely many shocks with possible collisions. The key ingredient in our approach is an interpolation inequality between the L 1 and Lip + -bounds, which enables us to convert a global result into a (non-optimal) local estimate. A bootstrap argument yields optimal pointwise error bound for both the vanishing viscosity and finite difference approximations. 1. Introduction We study solutions to the single hyperbolic conservation laws with small viscosity of the form u ffl t + f(u ffl ) x = fflu ffl xx ; x 2 R; t ? 0; ffl ? 0 (1) subject to the initial condition u ffl 0 (x) = u 0 (x): (2) We are interested in the relation between its solution, u ffl , and the solution u of the corresponding conservation laws without viscosity u t + f(u) x = 0; x 2 R; t ? 0: (3) The...

Abstract. 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 L1-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 L1-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. 1.

. We study the structure and smoothness of non-homogeneous convex conservation laws. We address the question regarding the number of smoothness pieces. We show that under certain conditions on the initial data the entropy solution has only finite number of discontinuous curves. We also obtain some global estimates on derivatives of the piecewisely 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. 1. Introduction We consider the initial value problem for the non-homogeneous scalar conservation law @ t u + @ x f(u) + g(u) = 0 (x; t) 2 R \Theta R + (1.1) subject to the initial condition u(x; 0) = u 0 (x) x 2 R; (1.2) where the flux f is strictly convex f 00 fl ? 0 (1.3) and g satisfies g(0) = 0 and a Lipschitz condition with a Lipschitz constant L (L ? 0), i.e., jg(u) \Gamma g(v)j Lju \Gamma vj: (1.4) In general, the problem (1.1)...

In this paper we address the questions of the convergence rate for approximate solutions to conservation laws with piecewise smooth solutions in a weighted W 1;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 W 1;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 W 1;1 -convergence theory is that for convex conservation laws we indeed have W 1;1 -error bounds for the approximate solutions to conservation laws. Furthermore, the O(ffl)-pointwise error estimates of Tadmor and Tang [SIAM J. Numer. Anal., 36 (1999), pp. 1739-1758] is recovered by the use of our W 1;1 -convergence theory. AMS(MOS) subject classification...

We consider the convergence and stability property of MUSCL relaxing schemes applied to conservation laws with stiff source terms. The maximum principle for the numerical schemes will be established. It will be also shown that the MUSCL relaxing schemes are uniformly l 1 - and TV -stable in the sense that they are bounded by a constant independent of the relaxation parameter ffl, the Lipschitz constant of the stiff source term and the time step \Deltat. The Lipschitz constant of the l 1 continuity in time for the MUSCL relaxing schemes is shown to be independent of ffl and \Deltat. The convergence of the relaxing schemes to the corresponding MUSCL relaxed schemes is then established.

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