A Note on the Numerical Solution of the Wave Equation With Piecewise Smooth Coefficients

The numerical solution of the initial value problem for the wave equation is considered for the case when the equation coefficients are piecewise smooth. This problem models linear wave propagation in a medium in which the properties of the medium change discontinuously at interfaces. Convergent difference approximations can be found that do not require the explicit specification of the boundary conditions at interfaces in the medium and hence are simple to program. Although such difference approximations typically can only be expected to be first-order accurate, the numerical phase velocity has the same accuracy as the difference approximation would if the coefficients in the differential equation were smoooth. This is proved for the one-dimensional case and demonstrated numerically for an example in two space dimensions in which the interface is not aligned with the computational mesh.