Order and Stiffness of the Gauss Runge-kutta Method for Initial-boundary Value Problems

Abstract. Existing analysis shows that when the Gauss Runge-Kutta (GRK) (also called Legendre-Gauss collocation) formulation with s Gaussian nodes is applied to ordinary differential equation initial value problems, the discretization has order 2s (super-convergent) [8]. However, for time-dependent partial differential equations (PDEs) with boundary conditions, super-convergence is only observed numerically for problems with periodic boundary conditions, and the numerical scheme’s order is s + k for general boundary conditions, where k ≥ 0 depends on the types of the boundary conditions and how they are imposed numerically. In this paper, we show how the GRK formulation for time-dependent PDEs can be solved accurately and efficiently using the Krylov deferred correction methods [11] and fast elliptic equation solvers, and analyze its order behaviors and “stiffness”. Numerical experiments are presented to validate the theoretical results.