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Mathematics of Computation

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Convergence rate of approximate solutions to weakly coupled nonlinear systems

Author: Haim Nessyahu
Journal: Math. Comp. 65 (1996), 575-586
MSC (1991): Primary 35L65; Secondary 65M10, 65M15
MathSciNet review: 1333321
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Abstract: We study the convergence rate of approximate solutions to nonlinear hyperbolic systems which are weakly coupled through linear source terms. Such weakly coupled $2 \times 2$ systems appear, for example, in the context of resonant waves in gas dynamics equations.

This work is an extension of our previous scalar analysis. This analysis asserts that a One Sided Lipschitz Condition (OSLC, or $\mathrm{Lip}^+$-stability) together with $W^{-1,1}$-consistency imply convergence to the unique entropy solution. Moreover, it provides sharp convergence rate estimates, both global (quantified in terms of the $W^{s,p}$-norms) and local.

We focus our attention on the $\mathrm{Lip}^+$-stability of the viscosity regularization associated with such weakly coupled systems. We derive sufficient conditions, interesting for their own sake, under which the viscosity (and hence the entropy) solutions are $\mathrm{Lip}^+$-stable in an appropriate sense. Equipped with this, we may apply the abovementioned convergence rate analysis to approximate solutions that share this type of $\mathrm{Lip}^+$-stability.

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Received by editor(s): May 24, 1993
Additional Notes: This research was supported in part by the Basic Research Foundation, Israel Academy of Sciences and Humanities
$^{1}$ Passed away in the dawn of April the 26th, 1994, at age 29, in The Himalayas, Nepal
Article copyright: © Copyright 1996 American Mathematical Society