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<p>We have found some rigorous discrete Green–Gauss laws among finite difference/volume operators based on the Voronoi decompositions and convex polygonal meshes. Some of those definitions and laws differ from those appearing in conventional studies. Furthermore, we can apply them to design structure-preserving numerical methods for some partial differential equation problems and run numerical computations. Here, we would like to indicate those finite difference/volume operators, Green–Gauss laws, and the obtained discrete variational derivative methods, one of the structure-preserving methods for partial differential equations, based on Voronoi cells. Furthermore, we expand the context to ones on arbitrary convex polygonal meshes. This expansion means we can design structure-preserving numerical schemes on arbitrary convex polygonal meshes.</p>

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Structure-preserving methods with difference operators on volume elements via Voronoi meshes and convex polygonal meshes

  • 期刊论文

  • 发表日期：

    2025年8月20日

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  • Daisuke Furihata

摘要

We have found some rigorous discrete Green–Gauss laws among finite difference/volume operators based on the Voronoi decompositions and convex polygonal meshes. Some of those definitions and laws differ from those appearing in conventional studies. Furthermore, we can apply them to design structure-preserving numerical methods for some partial differential equation problems and run numerical computations. Here, we would like to indicate those finite difference/volume operators, Green–Gauss laws, and the obtained discrete variational derivative methods, one of the structure-preserving methods for partial differential equations, based on Voronoi cells. Furthermore, we expand the context to ones on arbitrary convex polygonal meshes. This expansion means we can design structure-preserving numerical schemes on arbitrary convex polygonal meshes.