Stringent error estimates for one-dimensional, space-dependent $2\times2$ relaxation systems

Debora Amadori (1) , Laurent Gosse (2)

1 DISIM - Department of Information Engineering, Computer Science and Mathematics = Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica [L'Aquila]
2 IAC - Istituto per le Applicazioni del Calcolo "Mauro Picone"

Sharp and local $L^1$ {\it a-posteriori} error estimates are established for so--called "well-balanced" $BV$ (hence possibly discontinuous) numerical approximations of $2 \times 2$ space-dependent Jin-Xin relaxation systems under sub-characteristic condition. According to the strength of the relaxation process, one can distinguish between two complementary regimes: 1/ a weak relaxation, where local $L^1$ errors are shown to be of first order in $\DX$ and uniform in time, 2/ a strong one, where numerical solutions are kept close to entropy solutions of the reduced scalar conservation law, and for which Kuznetsov's theory indicates a behavior of the $L^1$ error in $t\cdot \sqrt{\DX}$. The uniformly first-order accuracy in weak relaxation regime is obtained by carefully studying interaction patterns and building up a seemingly original variant of Bressan-Liu-Yang's functional, able to handle $BV$ solutions of arbitrary size for these particular inhomogeneous systems. The complementary estimate in strong relaxation regime is proven by means of a suitable extension of methods based on entropy dissipation for space-dependent problems

Mots clés: Well-balanced scheme; Bressan-Liu-Yang functional; Entropy dissipation; Well-balanced scheme.; Kuznetsov's method; $L^1$ error estimate; position-dependent relaxation model

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Debora Amadori, Laurent Gosse. Stringent error estimates for one-dimensional, space-dependent $2\times2$ relaxation systems. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2015, pp.23. ⟨10.1016/j.anihpc.2015.01.001⟩. ⟨hal-01057737⟩