From Semidiscrete to Fully Discrete: Stability of Runge--Kutta Schemes by The Energy Method

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Abstract
The integration of semidiscrete approximations for time-dependent problems is encountered in a variety of applications. The Runge--Kutta (RK) methods are widely used to integrate the ODE systems which arise in this context, resulting in large ODE systems called methods of lines. These methods of lines are governed by possibly ill-conditioned systems with a growing dimension; consequently, the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading. Instead, we present here a stability analysis of RK methods for well-posed semidiscrete approximations, based on a general energy method. We review the stability question for such RK approximations, and highlight its intricate dependence on the growing dimension of the problem. In particular, we prove the strong stability of general fully discrete RK methods governed by coercive approximations. We conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection-diffusion equations with variable coefficients. A straightforward implementation of our results verify the strong stability of RK methods for local finite-difference schemes as well as global spectral approximations.Since our approach is based on the energy method (which is carried in the physical space), and since it avoids the von Neumann analysis (which is carried in the dual Fourier space), we are able to easily adapt additional extensions due to nonperiodic boundary conditions, general geometries, etc.