Stable, high-order discretization for evolution of the wave equation in 1 + 1 dimensions

We carry forward the approach of Alpert, Greengard, and Hagstrom to construct stable high-order explicit discretizations for the wave equation in one space and one time dimension. They presented their scheme as an integral form of the Lax–Wendroff method. Our perspective is somewhat different from theirs; our focus is on the discretization of the evolution formula rather than on its form (integral, differential, etc.). A key feature of our approach is the independent computation of three discretizations, one for bulk (away from boundaries) propagation, one for propagation near boundaries, and a projection operator to enforce boundary conditions. This is done in a way that is straightforward to extend to more space dimensions.