Method of Lines Transpose: An Implicit Solution to the One Dimensional Wave Equation

We present a new method for solving the wave equation implicitly in one spatial dimension. Our approach is to discretize the wave equation in time, following the method of lines transpose, sometimes referred to as the transverse method of lines, or Rothe's method. We then solve the resulting system of partial differential equations using boundary integral methods. Our algorithm extends to higher spatial dimensions using an alternating direction implicit (ADI) framework. Thus we develop a boundary integral solution that is competitive with explicit finite difference methods, both in terms of accuracy and speed. However, it provides more flexibility in the treatment of source functions and complex boundaries. We provide the analytical details of our one-dimensional method herein, along with a proof of the convergence of our schemes in free space and on a bounded domain. We find that the method is unconditionally stable and achieves second order accuracy. Upon examining the discretization error, we derive a novel optimal quadrature method, which can be viewed as a Lax-type correction.