Grady Wright
Department of Mathematics
University of Utah
Scattered Node Finite Difference-Type Formulas Generated from
Radial Basis Functions
Abstract
In the finite difference (FD) method for solving partial differential
equations (PDEs), derivatives at a node are approximated by a weighted sum
of function values at some surrounding nodes. In the one dimensional
case, the weights of the FD formulas are conveniently computed using
polynomial interpolation. These one dimensional formulas can be combined
to create FD formulas for partial derivatives in two and higher
dimensions. This strategy, however, requires that the nodes of the FD
"stencils" are situated on some kind of structured grid (or collection of
structured grids), which severely limits the application of the FD method
to PDEs in irregular geometries. In this talk, we present a novel
approach that resolves this problem by allowing the nodes of the FD
stencils to be placed freely and by using radial basis function (RBF)
interpolation for computing the corresponding weights in the scattered
node FD-type formulas. We show how this RBF approach can exactly
reproduce all classical FD formulas and how compact FD formulas can be
generalized to scattered nodes and RBFs. This latter result is important
in that it allows the number of nodes in the stencils to remain relatively
low without sacrificing accuracy. For the Poisson equation, these new
compact scattered node schemes can also be made diagonally dominant, which
ensures both a high degree of robustness and applicability of iterative
methods. We conclude the talk with some numerical examples and future
applications of the method.
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