Asymptotic-preserving finite difference method for partially dissipative hyperbolic systems

Abstract

We analyse the preservation of asymptotic properties of partially dissipative hyperbolic systems when switching to a fully discrete setting. We prove that one of the simplest consistent and unconditionally stable numerical methods—the implicit central finite-difference scheme—preserves both the large time asymptotic behaviour and the parabolic relaxation limit of one-dimensional partially dissipative hyperbolic systems that 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, in the continuous case, solutions are shown to exhibit distinct behaviour in low and high frequencies we introduce a novel discrete Littlewood–Paley decomposition tailored to the central finite-difference scheme. This allows us to prove Bernstein-type estimates for discrete differential operators and leads to new diffusive limit results such as the strong convergence of the discrete linearized compressible Euler system with damping towards the discrete heat equation, uniformly with respect to the spatial mesh parameter.