An efficient high-order meshless method for advection-diffusion equations on time-varying irregular domains

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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The study introduces a high-order radial basis function finite difference method for solving advection-diffusion equations on time-varying domains. The framework eliminates overlap parameters, enables tuning-free assembly of differentiation matrices on moving domains, and demonstrates high performance with high convergence rates.