Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation∗

We develop high-resolution shock-capturing numerical schemes for hyperbolic systems with relaxation. In such systems the relaxation time may vary from order-1 to much less than unity. When the relaxation time is small, the relaxation term becomes very strong and highly stiff, and underresolved numerical schemes may produce spurious results. Usually one cannot decouple the problem into separate regimes and handle different regimes with different methods. Thus it is important to have a scheme that works uniformly with respect to the relaxation time. Using the Broadwell model of the nonlinear Boltzmann equation we develop a second-order scheme that works effectively, with a fixed spatial and temporal discretization, for all ranges of the mean free path. Formal uniform consistency proof for a first-order scheme and numerical convergence proof for the second-order scheme are also presented. We also make numerical comparisons of the new scheme with some other schemes. This study is motivated by the reentry problem in hypersonic computations.