Local error analysis for approximate solutions of hyperbolic conservation laws
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Springer-Verlag 2005
Local error analysis for approximate solutions of hyperbolic conservation laws Smadar Karni a and Alexander Kurganov b a Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA
E-mail: [email protected] b Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
E-mail: [email protected]
Received 3 June 2002; accepted 17 March 2003 Communicated by E. Tadmor
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∞ loc estimates, following the Lip convergence theory developed by Tadmor et al. Comparisons between the local truncation error and the L∞ loc -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 in [Karni, Kurganov and Petrova, J. Comput. Phys. 178 (2002) 323–341]. Keywords: conservation laws, weak solutions, truncation error, error estimates AMS subject classification: 65M15, 35L65
1.
Introduction
We consider the system of one-dimensional hyperbolic conservation laws subject to the compactly supported (or periodic) initial data, ut + f (u)x = 0,
u(x, t = 0) = u0 (x).
(1.1)
Such systems arise in many different applications, including fluid mechanics, astrophysics, meteorology, semiconductors and others – thus, solving (1.1) is of a great practical importance. System (1.1) admits the (finite time) formation of shock discontinuities, even for infinitely smooth initial data. This makes the theory of (1.1) difficult and the design of numerical methods challenging (a detailed review of a variety of modern methods and approaches can be found in, e.g., [1,3,6,9]).
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S. Karni, A. Kurganov / Error analysis for conservation laws
Beyond the time of shock formation, classical solutions cease to exist and the solution of (1.1) is extended as a weak solution in the sense of distributions. Definition 1.1. u(x, t) is a weak solution of the IVP (1.1) if it satisfies ∞ u(x, t)φt (x, t)+f u(x, t) φx (x, t) dx dt+ u(x, 0)φ(x, 0) dx = 0 (1.2) − t =0
X
X
for all test-functions φ(x, t) ∈
C0∞ (X
× [0, ∞[).
Weak solutions are not unique and an additional criterion, known as an entropy condition, is used to select the unique physically relevant solution (see, e.g., [14]). In this paper, we focus on the problem of measuring local errors in numerical solutions. Since solutions of nonlinear conservation laws are generally nonsmooth, standard methods of convergence rate analysis, based on the Taylor expansions (see, e.g., [12]), are invalid. Hi
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