On the stability of two-derivative time discretizations

Abstract

In this paper, we analyze stability properties of the two-derivative strong stability preserving schemes presented in [Gottlieb et al., SIAM Journal on Numerical Analysis 60, 2022]. Stability analysis shows that the diagonally implicit two-derivative two-stage third-order strong stability preserving scheme can never be A-stable. We provide a detailed investigation of the third-order schemes and discuss stabilizing strategies. The stabilizing techniques are applicable to tune any general implicit two-derivative scheme. We implement the two-derivative strong stability preserving schemes for partial differential equations with a discontinuous Galerkin spectral element spatial discretization. We use Newton's method for non-linear stage equations and the generalized minimal residual method with a matrix-free approach for solving linear algebraic equations under suitable precondition-ing. The method is applied for compressible Euler and Navier-Stokes equations with orders up to four. Numerical results show that the second and fourth-order strong stability preserving schemes attain their desired order of convergence for relatively large timesteps. In contrast, third-order schemes require smaller timesteps to exhibit convergence. Nevertheless, the improved adaptive third-order scheme yields stable solutions.