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-differences 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 equations with damping towards the discrete heat equation, uniformly with respect to the mesh parameter.

翻译：本文分析了部分耗散双曲系统在离散化框架下的渐近性质保持问题。我们证明，对于满足卡尔曼秩条件的一维部分耗散双曲系统，一种最简单且具备一致性与无条件稳定性的数值方法——中心有限差分格式——能够同时保持其渐近行为与抛物松弛极限。通过基于低耗散理论构造时间加权扰动能量泛函，我们实现了长时间渐近保持性质。对于松弛保持性质，我们观察到连续情形下解在低频与高频区域呈现不同行为，受此启发，我们针对中心有限差分格式引入了一套新颖的离散Littlewood-Paley理论。该方法为离散微分算子证明了Bernstein型估计，并推导出新的松弛结果：带阻尼的离散线性化可压缩欧拉方程关于网格参数一致强收敛于离散热方程。