Stringent error estimates for one-dimensional, space-dependent 2×2 relaxation systems

International audienceSharp 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 problem