Trajectory-Based RBF Collocation Method for Surface Advection-Diffusion Equations

TL;DR

This paper presents a novel trajectory-based RBF collocation method for solving surface advection-diffusion equations, effectively decoupling advection and diffusion processes on manifolds with proven equivalence and demonstrated stability and accuracy.

Contribution

It introduces a new operator-split characteristic method using RBF collocation for surface PDEs, with rigorous proof of equivalence and an intrinsic approach for manifold embedding.

Findings

Method is robust and stable in numerical experiments

Achieves high accuracy without operator splitting error

Effectively handles surface advection-diffusion equations

Abstract

We introduce a Trajectory-Based RBF Collocation (TBRBF) method for solving surface advection-diffusion equations on smooth, compact manifolds. TBRBF decouples advection and diffusion by applying a characteristic treatment with a Kansa-type RBF collocation method for diffusion PDE, which yields an operator-split characteristic (OSC) system comprising a characteristic ODE and a diffusion PDE. We rigorously prove the equivalence between the OSC system and the original surface PDE on manifolds by embedding the latter into a narrow band domain. Using an intrinsic approach, we construct a time-continuous embedded PDE with push-forward operators in each chart of the atlas and establish its equivalence with the OSC system in the narrow band. Restricting the solution back to the manifold recovers the OSC system on manifolds, ensuring that the method introduces no operator splitting error. Extensive numerical experiments confirm the robust stability and accuracy of the proposed method.