November 11, 2009
Grady Wright, Wed. November 18, 2:40-3:30 pm, MG 120.
Scattered Node Finite Difference-Type Formulas Generated from Radial Basis Functions with Applications
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 for geophysical problems.