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arXiv · 2312.00327

A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains

Leticia Mattos Da Silva·Oded Stein·Justin Solomon
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Abstract

We introduce a framework for solving a class of parabolic partial differential equations on triangle mesh surfaces, including the Hamilton-Jacobi equation and the Fokker-Planck equation. PDE in this class often have nonlinear or stiff terms that cannot be resolved with standard methods on curved triangle meshes. To address this challenge, we leverage a splitting integrator combined with a convex optimization step to solve these PDE. Our machinery can be used to compute entropic approximation of optimal transport distances on geometric domains, overcoming the numerical limitations of the state-of-the-art method. In addition, we demonstrate the versatility of our method on a number of linear and nonlinear PDE that appear in diffusion and front propagation tasks in geometry processing.

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Leticia Mattos Da Silva, Oded Stein, Justin Solomon. 2024-06-02. A Framework for Solving Parabolic Partial Differential Equations on Discrete Domains. https://doi.org/10.1145/3666087

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