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 › Journal article › peer-review

26 Citations (Scopus)

34 Downloads (Pure)

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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author = "Eitan Tadmor and Tao Tang",

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AU - Tadmor, Eitan

AU - Tang, Tao

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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.

KW - Conservation laws

KW - Error estimates

KW - Maximum principle

KW - One-sided interpolation inequality

KW - Optimal convergence rate

KW - Relaxation method

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Pointwise Error Estimates for Relaxation Approximations to Conservation Laws

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