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Local error analysis for approximate solutions of hyperbolic conservation laws. (English) Zbl 1127.65070
Summary: We consider approximate solutions to nonlinear hyperbolic conservation laws. If the exact solution is unavailable, the truncation error may be the only quantitative measure for the quality of the approximation. We propose a new way of estimating the local truncation error, through the use of localized test-functions. In the convex scalar case, they can be converted into \(L_{\text{loc}}^{\infty}\) estimates following the \(Lip'\) convergence theory developed by E. Tadmor and T. Tang [SIAM J. Numer. Anal. 36, No. 6, 1739–1758 (1999; Zbl 0934.35088)]. Comparisons between the local truncation error and the \(L_{\text{loc}}^{\infty}\)-error show remarkably similar behavior.
Numerical results are presented for the convex scalar case, where the theory is valid, as well as for nonconvex scalar examples and the Euler equations of gas dynamics. The local truncation error has proved a reliable smoothness indicator and has been implemented in adaptive algorithms by the authors and G. Petrova [J. Comput. Phys. 178, 323–341 (2002; Zbl 0998.65092)].

MSC:
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
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