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.

翻译：我们提出了一种在三角形网格曲面上求解一类抛物型偏微分方程的框架，包括 Hamilton-Jacobi 方程和 Fokker-Planck 方程。该类中的偏微分方程通常包含非线性或刚性项，无法通过标准方法在弯曲三角形网格上求解。为解决这一挑战，我们采用了一种结合凸优化步骤的分裂积分器来求解这些偏微分方程。我们的工具可用于计算几何域上最优传输距离的熵近似，克服了现有最先进方法的数值局限性。此外，我们通过几何处理中扩散和波前传播任务中出现的多个线性和非线性偏微分方程，展示了本方法的通用性。