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Description: TITLE: HIGH-ORDER AND GEOMETRICALLY FLEXIBLE NUMERICAL METHOD FOR SYSTEM OF COUPLED TIME-DEPENDENT PARTIAL DIFFERENTIAL EQUATIONS
Abstract:Systems of partial differential equations (PDEs) are often encountered in many scientific applications. One particular example related to mathematical biology is a system of coupled, nonlinear reaction diffusion PDEs that models the spread of harmful plaques associated with Alzheimer's disease on the human brain. Numerical collocation methods for computing and simulating solutions for such problems face several challenges including dealing with irregular geometry, capturing local profiles, handling non-linearity, and maintaining stability, efficiency, and accuracy. In our preliminary work, we numerically study this type of system on regular domains using method of lines, where classical finite difference (FD) is used for the spatial discretization and coupled with an Implicit-Explicit (IMEX) time stepping scheme to advance the solution in time. We have solved a system of up to 50 coupled PDEs and tested for convergence in space and time using known solutions. Our next goal, which is the scope of this proposal, is to explore a new approach using Radial Basis Function in finite difference mode (RBF-FD) methods for the spatial discretization that can mitigate drawbacks arising from using FD methods on complex domains. Due to the nature of collocation methods, implementing RBF-FD methods will require minimal modifications to the existing model. We plan to study and address issues that arise from the implementation of RBF-FD methods when coupled with IMEX time integrators on regular and irregular domains with boundary conditions. We will use a variety of numerical techniques to resolve any issues that may arise from this combination of spatial/temporal numerical schemes such as instability, dealing with boundary conditions, preconditioning techniques for solving large systems, and resolving local profiles. Additionally, we will place emphasis on reproducible research codes which will be made publicly available through a git repository.