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.

翻译：本文分析了部分耗散双曲系统在离散化过程中渐近性质的保持性。我们证明，满足卡尔曼秩条件的一维部分耗散双曲系统，即便采用最简单的一致且无条件稳定的数值方法——中心有限差分格式——也能同时保持渐近行为及抛物松弛极限。通过基于低共轭性理论构造时变加权扰动能量泛函，实现了大时间尺度下的渐近保持特性。针对松弛保持性质，受连续情形中解在低高频呈现不同行为的启发，我们创新性地提出了适配中心有限差分格式的离散Littlewood-Paley理论。该理论可证明离散微分算子的Bernstein型估计，并导出新的松弛结果：带阻尼的离散线性化可压欧拉系统在网格参数一致意义下强收敛于离散热方程。