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High-order asymptotic-preserving methods for fully nonlinear relaxation problems. (English) Zbl 1426.76455

Summary: We study solutions to nonlinear hyperbolic systems with fully nonlinear relaxation terms in the limit of, both, infinitely stiff relaxation and arbitrary late time. In this limit, the dynamics is governed by effective systems of parabolic-type which may contain degenerate and/or fully nonlinear diffusion terms. For this class of problems, we develop an implicit-explicit method based on Runge–Kutta discretization in time, and we apply this method to the investigation of several examples of interest in fluid dynamics. Importantly, we impose here a realistic stability condition on the time step and we demonstrate that solutions in the hyperbolic-to-parabolic regime can be computed numerically with high robustness and accuracy, even in the presence of fully nonlinear relaxation.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
35L75 Higher-order nonlinear hyperbolic equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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