Abstract
We propose to use finite elements and BDF2 time stepping to solve the problem of computing a solution to the time dependent wave equation with a variable sound speed in an infinite sound hard pipe (waveguide). By using the Laplace transform and an appropriate Dirichlet-to-Neumann (DtN) map for the problem, we can prove that this problem can be reduced to a variational problem on a bounded domain that has a unique solution. This solution can be discretized in space using finite elements (projecting into a Fourier space on the two artificial boundaries to allow the rapid calculation of the DtN map). We discretize in time using the Convolution Quadrature (CQ) approach and in particular BDF2 time-stepping. Thanks to CQ we obtain a stable and convergent discretization of the DtN map, and hence of the fully discrete BDF2-finite element scheme without a CFL condition. We illustrate the method with some numerical results.
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