Loss of High Order Spatial Accuracy Due to Boundary Error Caused by Runge-Kutta Time Integration

Explicit Runge–Kutta methods have frequently been used for solving initial boundary value problems with the method of lines. For linear and certain non-linear problems like the inviscid Burgers’ equation, the correct specification of Dirichlet boundary conditions at the intermediate Runge–Kutta stages can be derived analytically. For general non-linear PDEs and general boundary conditions, it is currently not known how to find consistent analytical boundary conditions that do not lower the formal accuracy of the scheme. There are some numerical approaches that gain full accuracy but lead to deteriorated stability conditions. Here we focus on solving non-linear wave like equations using high-order finite difference methods. We examine the properties of an inconsistent boundary treatment and make comparisons with a correct one when applicable. We examine the effect of introducing viscosity. We contrast fourth order Runge–Kutta and Adams–Bashforth time integrators