From semidiscrete to fully discrete: stability of Runge-Kutta schemes by the energy method. II

We study the stability of Runge-Kutta methods for the time integration of semidiscrete systems associated with time dependent PDEs. These semidiscrete systems amount to large systems of ODEs with the possibility that the matrices involved are far from being normal. The stability question of their Runge-Kutta methods, therefore, cannot be addressed by the familiar scalar arguments of eigenvalues lying in the corresponding region of absolute stability. Instead, we replace this scalar spectral analysis by the energy method, where stability of the fully discrete Runge-Kutta methods takes into account the full eigenstructure of the problem at hand. We discuss two energy method approaches that guarantee the stability of fullydiscrete Runge-Kutta methods for sufficiently small CFL condition, ∆t ≤ ∆t0. In the first approach, Runge-Kutta methods are shown to preserve stability for the subclass of coercive semidiscrete problems. A second approach treats the more general class of semibounded problems. It is shown that their time integration by third-order Runge-Kutta method is stable under a slightly more restrictive CFL condition. We conclude by utilizing these two approaches to examine the stability of Runge-Kutta discretizations of semidiscrete advection-diffusion problems. Our study includes a detailed stability analysis for prototype examples of one-sided and centered finite differencing and pseudospectral Fourier and Jacobi-based methods. ∗Department of Mathematics, UCLA, Los-Angeles CA 90095, USA. email: [email protected]. Research was supported in part by NSF grants DMS01-07428, DMS01-07917 and ONR grant N00014-91-J-1076.