Approximating moving point sources in hyperbolic partial differential equations

TL;DR

This paper develops a new discretization method for moving point sources in hyperbolic PDEs, ensuring accurate and stable numerical solutions by preventing mode excitation and spectrum dependence on source position.

Contribution

The authors introduce a novel source discretization technique for moving sources in hyperbolic equations, proving convergence and demonstrating effectiveness in 1D and 2D simulations.

Findings

Achieves design-order convergence for 1D advection equations.

Numerical experiments show effective convergence for 2D acoustic wave equations.

Discretization covers approximately √N grid points, suitable for non-boundary touching trajectories.

Abstract

We consider point sources in hyperbolic equations discretized by finite differences. If the source is stationary, appropriate source discretization has been shown to preserve the accuracy of the finite difference method. Moving point sources, however, pose two challenges that do not appear in the stationary case. First, the discrete source must not excite modes that propagate with the source velocity. Second, the discrete source spectrum amplitude must be independent of the source position. We derive a source discretization that meets these requirements and prove design-order convergence of the numerical solution for the one-dimensional advection equation. Numerical experiments indicate design-order convergence also for the acoustic wave equation in two dimensions. The source discretization covers on the order of $\sqrt{N}$ grid points on an $N$-point grid and is applicable for source trajectories that do not touch domain boundaries.