The Correct Formulation of Intermediate Boundary Conditions for Runge-Kutta Time Integration of Initial Boundary Value Problems

Pseudospectral and high-order finite difference methods are well established for solving time-dependent partial differential equations by the method of lines. The use of highorder spatial discretizations has led in turn to a concomitant interest in high-order time stepping schemes, so that the temporal and spatial errors are of comparable magnitude. Explicit Runge-Kutta methods are widely used in practice, but a difficulty encountered with these is the loss of accuracy that results from wrong specifications of intermediate-stage boundary conditions. The best prescriptions to date can do no better than achieve thirdorder accuracy for general nonlinear problems. On the other hand, if these artificial boundary values are not explicitly prescribed but are computed by integrating the semi-discrete equations at the boundary, the maximum allowable time step is significantly reduced. The remedy proposed here is to prescribe analytically those values that would result from applying the Runge-Kutta solver at the boundaries, and hence maintain accuracy without incurring further step size restrictions. We describe in detail the implementation for hyperbolic equations, and present both scalar and vector examples.