Method of lines transpose: an efficient A-stable solver for wave propagation

TL;DR

This paper introduces an efficient, second-order, A-stable wave equation solver using the method of lines transpose, capable of handling various boundary conditions and suitable for coupling with particle codes.

Contribution

It develops a boundary condition implementation for the MOL$^T$ wave solver that maintains stability and accuracy, including outflow, Dirichlet, and Neumann conditions in 1D and 2D.

Findings

The scheme is A-stable and second-order accurate.

Boundary conditions are embedded without stability loss.

The solver effectively handles point and soft sources.

Abstract

Building upon recent results obtained in [7,8,9], we describe an efficient second order, A-stable scheme for solving the wave equation, based on the method of lines transpose (MOL$^T$), and the resulting semi-discrete (i.e. continuous in space) boundary value problem. In [7], A-stable schemes of high order were derived, and in [9] a high order, fast $\mathcal{O}(N)$ spatial solver was derived, which is matrix-free and is based on dimensional-splitting. In this work, are interested in building a wave solver, and our main concern is the development of boundary conditions. We demonstrate all desired boundary conditions for a wave solver, including outflow boundary conditions, in 1D and 2D. The scheme works in a logically Cartesian fashion, and the boundary points are embedded into the regular mesh, without incurring stability restrictions, so that boundary conditions are imposed without any reduction in the order of accuracy. We demonstrate how the embedded boundary approach works in the cases of Dirichlet and Neumann boundary conditions. Further, we develop outflow and periodic boundary conditions for the MOL$^T$ formulation. Our solver is designed to couple with particle codes, and so special attention is also paid to the implementation of point sources, and soft sources which can be used to launch waves into waveguides.