An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains

TL;DR

This paper introduces a high-order, tuning-free meshless RBF-FD method for efficiently solving advection-diffusion equations on complex, moving domains with demonstrated high accuracy and optimal computational complexity.

Contribution

The paper develops a novel automatic weight computation procedure for RBF-FD that removes the need for overlap parameters, enabling efficient high-order solutions on time-varying domains.

Findings

Achieves high-order convergence on 2D and 3D moving domains.

Demonstrates $O(N \,\log N)$ computational complexity.

Successfully applies to complex coupled 3D problems.

Abstract

We present a high-order radial basis function finite difference (RBF-FD) framework for the solution of advection-diffusion equations on time-varying domains. Our framework is based on a generalization of the recently developed Overlapped RBF-FD method that utilizes a novel automatic procedure for computing RBF-FD weights on stencils in variable-sized regions around stencil centers. This procedure eliminates the overlap parameter $\delta$, thereby enabling tuning-free assembly of RBF-FD differentiation matrices on moving domains. In addition, our framework utilizes a simple and efficient procedure for updating differentiation matrices on moving domains tiled by node sets of time-varying cardinality. Finally, advection-diffusion in time-varying domains is handled through a combination of rapid node set modification, a new high-order semi-Lagrangian method that utilizes the new tuning-free overlapped RBF-FD method, and a high-order time-integration method. The resulting framework has no tuning parameters and has $O(N \log N)$ time complexity. We demonstrate high-orders of convergence for advection-diffusion equations on time-varying 2D and 3D domains for both small and large Peclet numbers. We also present timings that verify our complexity estimates. Finally, we utilize our method to solve a coupled 3D problem motivated by models of platelet aggregation and coagulation, once again demonstrating high-order convergence rates on a moving domain.