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High-order asymptotic-preserving methods for fully nonlinear relaxation problems

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, with possibly degenerate and/or fully nonlinear diffusion terms. For this class of problems, we develop here an implicit-explicit method based on Runge-Kutta discretization in time, and we use this method in order to investigate several examples of interest in compressible 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 presence of fully nonlinear relaxation terms.