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arXiv · 2312.17478

Structure-preserving Kernel-based methods for solving dissipative PDEs on surfaces

Zhengjie Sun·Leevan Ling·Meng Chen
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Abstract

In this paper, we propose a general meshless structure-preserving Galerkin method for solving dissipative PDEs on surfaces. By posing the PDE in the variational formulation and simulating the solution in the finite-dimensional approximation space spanned by (local) Lagrange functions generated with positive definite kernels, we obtain a semi-discrete Galerkin equation that inherits the energy dissipation property. The fully-discrete structure-preserving scheme is derived with the average vector field method. We provide a convergence analysis of the proposed method for the Allen-Cahn equation. The numerical experiments also verify the theoretical analysis including the convergence order and structure-preserving properties.

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Zhengjie Sun, Leevan Ling, Meng Chen. 2024-12-18. Structure-preserving Kernel-based methods for solving dissipative PDEs on surfaces. https://arxiv.org/abs/2312.17478

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