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<p>This paper proposes the third-order, fifth-order, and seventh-order finite difference ghost multi-resolution weighted essentially non-oscillatory (GMR-WENO) schemes for solving Hamilton-Jacobi equations in one and two dimensions. It only utilizes the information defined on one four-point, one six-point, or one eight-point spatial stencil for designing high-order spatial approximations without introducing any other smaller stencils. The GMR-WENO schemes employ the orthogonal Legendre basis to design high degree reconstruction polynomials on these big spatial stencils. In addition, it uses the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2787_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2 \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> projection technique to derive a series of ghost low degree polynomials whose degree can gradually change from the highest degree to the zeroth degree. Some linear weights correlated to the real and ghost reconstruction polynomials may be any positive numbers, provided that their total sum equals one. Generally speaking, only one spatial stencil has been employed to design the real and ghost reconstruction polynomials and then devise high-order finite difference WENO schemes for simulating Hamilton-Jacobi equations on structured grids.</p>

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A New Type of Increasingly Higher Order of Finite Difference Ghost Multi-resolution WENO Schemes for Hamilton-Jacobi Equations

  • 期刊论文

  • 发表日期：

    2025年1月14日

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  • Yan Zhang,
  • Jun Zhu

摘要

This paper proposes the third-order, fifth-order, and seventh-order finite difference ghost multi-resolution weighted essentially non-oscillatory (GMR-WENO) schemes for solving Hamilton-Jacobi equations in one and two dimensions. It only utilizes the information defined on one four-point, one six-point, or one eight-point spatial stencil for designing high-order spatial approximations without introducing any other smaller stencils. The GMR-WENO schemes employ the orthogonal Legendre basis to design high degree reconstruction polynomials on these big spatial stencils. In addition, it uses the \(L^2 \) L 2 projection technique to derive a series of ghost low degree polynomials whose degree can gradually change from the highest degree to the zeroth degree. Some linear weights correlated to the real and ghost reconstruction polynomials may be any positive numbers, provided that their total sum equals one. Generally speaking, only one spatial stencil has been employed to design the real and ghost reconstruction polynomials and then devise high-order finite difference WENO schemes for simulating Hamilton-Jacobi equations on structured grids.