THE ODE FORMULATION OF HYPERBOLIC PDES DISCRETIZED BY THE SPECTRAL COLLOCATION METHOD

We formulate systems of hyperbolic partial differential equations (PDEs) with characteristic boundary conditions as pure initial value problems of ordinary differential equations (ODEs). This enables a straightforward application of any explicit or implicit method for the time integration of the spatially discretized system. We also show how this method relates to the correctional method of treating the boundary points. To exemplify, we work out the ODE formulation for the one-dimensional compressible Euler equations and a simple two-dimensional linear system and conclude with some numerical calculations on these, using both explicit and implicit Runge-Kutta methods.