Uniformly accurate schemes for hyperbolic systems with relaxation. (English) Zbl 0868.35070

From the authors’ abstract: The paper describes the development of high-resolution shock-capturing numerical schemes for hyperbolic systems with relaxation. 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, a second-order scheme is developed that works efficiently, 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. Furthermore, numerical comparisons of the new scheme with some other schemes are presented.


35L65 Hyperbolic conservation laws
34E13 Multiple scale methods for ordinary differential equations
49M25 Discrete approximations in optimal control
82B40 Kinetic theory of gases in equilibrium statistical mechanics


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