Asymptotic-preserving finite difference method for partially dissipative hyperbolic systems
<|reference_start|>Asymptotic-preserving finite difference method for partially dissipative hyperbolic systems: In this paper, we analyze the preservation of asymptotic properties of partially dissipative hyperbolic systems when switching to a discrete setting. We prove that one of the simplest consistent and unconditionally stable numerical methods - the central finite difference scheme - preserves both the asymptotic behaviour and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems which satisfy the Kalman rank condition.   The large time asymptotic-preserving property is achieved by conceiving time-weighted perturbed energy functionals in the spirit of the hypocoercivity theory. For the relaxation-preserving property, drawing inspiration from the observation that solutions in the continuous case exhibit distinct behaviours in low and high frequencies, we introduce a novel discrete Littlewood-Paley theory tailored to the central finite difference scheme. This allows us to prove Bernstein-type estimates for discrete differential operators and leads to a new relaxation result: the strong convergence of the discrete linearized compressible Euler system with damping towards the discrete heat equation, uniformly with respect to the mesh parameter.<|reference_end|>
@article{crin-barat2024asymptotic-preserving,
    title={Asymptotic-preserving finite difference method for partially dissipative
  hyperbolic systems},
    author={Timoth'ee Crin-Barat and Dragoc{s} Manea},
    journal={arXiv preprint arXiv:2404.06380},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.06380},
    primaryClass={math.AP cs.NA math.NA}
}