Symbolic Computation and Boundary Conditions for the Wave Equation

Abstract

We develop a procedure using Maple for the estimation of the accuracy of approximate boundary conditions for the wave equation. To solve this equation in an infinite three dimensional domain, a spherical artificial boundary is introduced to restrict the computational domain Ω. To determine the nonreflecting boundary condition on ∂Ω, we start with a finite number of spherical harmonics for the Helmholtz equation. With a precise choice of nodes on the sphere, the theorem on Gauss-Jordan quadrature establishes the discrete orthogonality of the spherical harmonics when summed over these nodes. The nonreflecting boundary condition for the Helmholtz equation follows readily upon solving the exterior Dirichlet problem. The boundary condition for the time dependent wave equation follows directly by taking the inverse Fourier transform of the boundary condition for the Helmholtz equation. The boundary condition has the following properties: only the first derivatives in space and time appear; once the coefficients are updated in a simple way from the previous time step, the boundary condition involves only the nodes at the current time step. We derive using Maple some very precise estimates of the accuracy of these approximate boundary conditions.