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<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2834_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="37" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sqrt{N}\)</EquationSource> <EquationSource Format="MATHML"><math> <msqrt> <mi>N</mi> </msqrt> </math></EquationSource> </InlineEquation> grid points on an <i>N</i>-point grid and is applicable for source trajectories that do not touch domain boundaries.</p>

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Approximating Moving Point Sources in Hyperbolic Partial Differential Equations

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年3月21日

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  • Ylva Ljungberg Rydin,
  • Martin Almquist

摘要

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 \(\sqrt{N}\) N grid points on an N-point grid and is applicable for source trajectories that do not touch domain boundaries.