We present a numerical method for computing the wave equation implicitly. The approach discretizes the wave equation in time using the method of lines transpose, also known as the transverse method of lines or Rothe's method. This differs from conventional methods in that we solve the resulting system of ODEs using boundary integral methods. We then analyze the fully discretized solution resulting from the midpoint and trapezoidal quadrature rules and show that convergent and unconditionally stable schemes result. We also show that the choice of discretization in time can lead to various schemes of prescribed accuracy, which may or may not introduce numerical dissipation into the approximate solution. We start with the simplest case of solving the wave equation in one dimension using either a free space solution or Dirichlet or Neumann Boundary conditions. We then analyze the stability and consistency of the method, as well as investigating the dispersion relations and deriving the phase error. Next, some numerical examples are presented which give validation to the error estimates. Further, the method is adapted for both outflow boundary conditions, using either one-way waves or a perfectly matched absorbing layer, as well as the implementation of a soft source. Finally, we utilize an ADI scheme to explore solving the wave equation in higher dimensions.Since this method of lines transpose approach is implicit, it removes the usual CFL stability limit inherent in explicit time stepping methods for solving the wave equation, and thus the algorithm will be more efficient than these explicit methods.
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