Stringent error estimates for one-dimensional, space-dependent 2x2 relaxation systems

Sharp and local L1 a-posteriori error estimates are established for so–called ”well- balanced” BV (hence possibly discontinuous) numerical approximations of 2 × 2 space-dependent Jin-Xin relaxation systems under sub-characteristic condition. Ac- cording to the strength of the relaxation process, one can distinguish between two complementary regimes: 1/ a weak relaxation, where local L1 errors are shown to be of first order in ∆x and uniform in time, 2/ a strong one, where numerical so- lutions are kept close to entropy solutions of the reduced scalar conservation law, and for which Kuznetsov’s theory indicates a behavior of the L1 error in t · √∆x. The uniformly first-order accuracy in weak relaxation regime is obtained by care- fully 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 re- laxation regime is proven by means of a suitable extension of methods based on entropy dissipation for space-dependent problems.