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Combining radial basis functions with the partition-of-unity method for numerically solving PDEs on the sphere
Abstract
We discuss a new collocation-type method for numerically solving partial differential equations (PDEs) on the sphere. The method uses radial basis function (RBF) approximations in a partition of unity framework for approximating spatial derivatives on the sphere. High-orders of accuracy are achieved for smooth solutions, while the overall computational cost of the method scales linearly with the number of unknowns. The discussion will be primarily limited to the transport equation and results will be presented for a few well-known test cases. We conclude with a preliminary application to the non-linear shallow water wave equations on a rotating sphere.
Approximation on surfaces with radial basis functions: from global to local methods
Abstract
Radial basis function (RBF) methods are becoming increasingly popular for numerically solving partial differential equations (PDEs) because they are geometrically flexible, algorithmically accessible, and can be highly accurate. There have been many successful applications of these techniques to various types of PDEs defined on planar regions in two and higher dimensions, and to PDEs defined on the surface of a sphere. Originally, these methods were based on global approximations and their computational cost was quite high. Recent efforts have focused on reducing the computational cost by using ``local’’ techniques, such as RBF generated finite differences (RBF-FD).
In this talk, we first describe our recent work on developing a new, high-order, global RBF method for numerically solving PDEs on relatively general surfaces, with a specific focus on reaction-diffusion equations. The method is quite flexible, only requiring a set of ``scattered’’ nodes on the surface and the corresponding normal vectors to the surface at these nodes. We next present a new scalable local method based on the RBF-FD approach with this same flexibility. This is the first application of the RBF-FD method to general surfaces. We conclude with applications of these methods to some biologically relevant problems.
This talk represents joint work with Edward Fuselier (High Point University), Aaron Fogelson, Mike Kirby, and Varun Shankar (all at the University of Utah).
A locally adaptive Cartesian finite-volume framework for solving PDEs on surfaces
Abstract
We describe our current efforts to develop finite volume
schemes for solving PDEs on logically Cartesian locally adapted
surfaces meshes. Our methods require an underlying smooth or
piecewise smooth grid transformation from a Cartesian computational
space to 3d surface meshes, but does not rely on analytic metric terms
to obtain second order accuracy. Our hyperbolic solvers are based on
Clawpack (R. J. LeVeque) and the parabolic solvers are based on a
diamond-cell approach (Y. Coudi\`ere, T. Gallou\"et, R. Herbin et
al). If time permits, I will also discuss Discrete Duality Finite
Volume methods for solving elliptic PDEs on surfaces.
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To do local adaption and time subcycling in regions requiring high
spatial resolution, we are developing ForestClaw, a hybrid adaptive
mesh refinement (AMR) code in which non-overlapping fixed-size
Cartesian grids are stored as leaves in a forest of quad- or
oct-trees. The tree-based code p4est (C. Burstedde) manages the
multi-block connectivity and is highly scalable in realistic
applications.
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I will present results from reaction-diffusion systems on surface
meshes, and test problems from the atmospheric sciences community.