The Theoretical Accuracy of Runge-Kutta Time Discretizations for the Initial Boundary Value Problem: A Careful Study of the Boundary Error
Abstract
The conventional method of imposing time dependent boundary conditions for Runge-Kutta (RK) 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 counter intuitive 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 RK cycle, (2) impose consistent intermediate boundary conditions derived from the physical boundary condition and its derivatives. The first method, while retaining the RK accuracy in all cases, results in a scheme with much reduced CFL condition, rendering the RK 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 non-linear hyperbolic equations the second method is effective only for for RK schemes of third order accuracy or less. Numerical studies are presented to verify the efficacy of each approach. Runge-Kutta schemes, Boundary conditions, Non-linear hyperbolic problems, High order schemes
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1993
- Accession Number
- ADA274824
Entities
People
- David Gottlieb
- Mark H. Carpenter
- Saul Abarbanel
- Wai-sun Don