Pointwise Error Estimates for Relaxation Approximations to Conservation Laws

Eitan Tadmor; Tao Tang

doi:10.1137/S0036141098349492

Pointwise Error Estimates for Relaxation Approximations to Conservation Laws

Eitan Tadmor^*, Tao Tang

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

25 Citations (Scopus)

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Abstract

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). A one-sided interpolation inequality between classical L¹ error estimates and Lip⁺ stability bounds enables us to convert a global L¹ result into a (nonoptimal) 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.

Original language	English
Pages (from-to)	870-886
Number of pages	17
Journal	SIAM Journal on Mathematical Analysis
Volume	32
Issue number	4
DOIs	https://doi.org/10.1137/S0036141098349492
Publication status	Published - Jul 2000

Scopus Subject Areas

Analysis
Computational Mathematics
Applied Mathematics

User-Defined Keywords

Conservation laws
Error estimates
Maximum principle
One-sided interpolation inequality
Optimal convergence rate
Relaxation method

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title = "Pointwise Error Estimates for Relaxation Approximations to Conservation Laws",

abstract = "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). A one-sided interpolation inequality between classical L1 error estimates and Lip+ stability bounds enables us to convert a global L1 result into a (nonoptimal) 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.",

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N2 - 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). A one-sided interpolation inequality between classical L1 error estimates and Lip+ stability bounds enables us to convert a global L1 result into a (nonoptimal) 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.

AB - 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). A one-sided interpolation inequality between classical L1 error estimates and Lip+ stability bounds enables us to convert a global L1 result into a (nonoptimal) 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.

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KW - Error estimates

KW - Maximum principle

KW - One-sided interpolation inequality

KW - Optimal convergence rate

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Pointwise Error Estimates for Relaxation Approximations to Conservation Laws

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