The Theoretical Accuracy of Runge–Kutta Time Discretizations for the Initial Boundary Value Problem: A Study of the Boundary Error

Abstract

The conventional method of imposing time-dependent boundary conditions for Runge–Kutta time advancement reduces the formal accuracy of the space-time method to first-order locally, and second-order globally, independently of the spatial operator. This counterintuitive result is analyzed in this paper. Two methods of eliminating this problem are proposed for the linear constant coefficient case. 1. Impose the exact boundary condition only at the end of the complete Runge–Kutta cycle. 2. Impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the Runge–Kutta accuracy in all cases, results in a scheme with a much reduced CFL condition, rendering the Runge–Kuua scheme less attractive. The second method retains the same allowable time step as the periodic problem. However, it is a general remedy only for the linear case. For nonlinear hyperbolic equations the second method is effective only for Runge–Kutta schemes of third-order accuracy or less. Numerical studies are presented to verify the efficacy of each approach.