An embedded method-of-lines approach to solving partial differential equations on surfaces

2013/07/22 by Ingrid von Glehn, von Glehn, Ingrid, Thomas März +3 · 2 citations
Engineering · Physics and Astronomy · #58J35 #65M06 #65M20 #Advanced Numerical Methods in Computational Mathematics #Electromagnetic Scattering and Analysis #Electromagnetic Simulation and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1307.5657

openalex publication_date 2013/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a method-of-lines formulation of the closest point method, a numerical technique for solving partial differential equations (PDEs) defined on surfaces. This is an embedding method, which uses an implicit representation of the surface in a band containing the surface. We define a modified equation in the band, obtained in a straightforward way from the original evolution PDE, and show that the solutions of this equation are consistent with those of the surface equation. The resulting system can then be solved with standard implicit or explicit time-stepping schemes, and the solutions in the band can be restricted to the surface. Our derivation generalizes existing formulations of the closest point method and is amenable to standard convergence analysis.

Citations

• The Closest Point Method and multigrid solvers for elliptic equations on surfaces

Cited by

• The Closest Point Method for Surface PDEs with General Boundary Conditions
• The Closest Point Heat Method for Solving Eikonal Equations on Implicit Surfaces

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