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 δ, 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.
This is an author-produced, peer-reviewed version of this article. © 2021, Elsevier. Licensed under the Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International license. The final, definitive version of this document can be found online at Journal of Computational Physics, https://doi.org/10.1016/j.jcp.2021.110633
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This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.
Shankar, Varun; Wright, Grady B.; and Fogelson, Aaron L. (2021). "An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains". Journal of Computational Physics, 445, 110633. https://doi.org/10.1016/j.jcp.2021.110633