THE THEORETICAL ACCURACY OF RUNGE-KUTTA TIME DISCRETIZATIONS FOR THE INITIAL-BOUNDARY VALUE-PROBLEM - STUDY OF THE BOUNDARY ERROR

The conventional method of imposing time-dependent boundary conditions for Runge-Kutta time advancement reduces the formal accuracy of the s pace-time method to first-order locally, and second-order globally, in dependently of the spatial operator. This counterintuitive result is a nalyzed in this paper. Two methods of eliminating this problem are pro posed for the linear constant coefficient case. 1. Impose the exact bo undary condition only at the end of the complete Runge-Kutta cycle. 2. Impose consistent intermediate boundary conditions derived from the p hysical boundary condition and its derivatives. The first method, whil e retaining the Runge-Kutta accuracy in all cases, results in a scheme with a much reduced cn condition, rendering the Runge-Kutta scheme le ss 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 e ffective only for Runge-Kutta schemes of third-order accuracy or less. Numerical studies are presented to verify the efficacy of each approa ch.