TY - JOUR
T1 - An Efficient High-Order Meshless Method for Advection-Diffusion Equations on Time-Varying Irregular Domains
AU - Shankar, Varun
AU - Wright, Grady B.
AU - Fogelson, Aaron L.
N1 - Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/11/15
Y1 - 2021/11/15
N2 - 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.
AB - 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.
KW - RBF-FD
KW - advection-diffusion
KW - high-order method
KW - meshfree
KW - radial basis function
KW - semi-Lagrangian
UR - http://www.scopus.com/inward/record.url?scp=85113353949&partnerID=8YFLogxK
UR - https://scholarworks.boisestate.edu/math_facpubs/256
U2 - 10.1016/j.jcp.2021.110633
DO - 10.1016/j.jcp.2021.110633
M3 - Article
SN - 0021-9991
VL - 445
JO - Journal of Computational Physics
JF - Journal of Computational Physics
M1 - 110633
ER -