A Meshfree Generalized Finite Difference Method for Surface PDEs
| Autor: | Pratik Suchde, Joerg Kuhnert |
|---|---|
| Přispěvatelé: | Publica |
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Surface (mathematics)
Finite difference method Finite difference Surface gradient FOS: Physical sciences Numerical Analysis (math.NA) 010103 numerical & computational mathematics Computational Physics (physics.comp-ph) 01 natural sciences Manifold 010101 applied mathematics Computational Mathematics Computational Theory and Mathematics Modeling and Simulation Metric (mathematics) FOS: Mathematics Tangent space Applied mathematics Mathematics - Numerical Analysis 0101 mathematics Laplace operator Physics - Computational Physics Mathematics |
| Popis: | In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to some existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinuous surface properties with large jumps are also introduced, and a few practical applications are presented. |
| Databáze: | OpenAIRE |
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