Eigenvalue stability of radial basis function discretizations for time-dependent problems

Date
2005
Authors
Platte, R.B.
Driscoll, Tobin A.
Journal Title
Journal ISSN
Volume Title
Publisher
Department of Mathematical Sciences
Abstract
Differentiation matrices obtained with infinitely smooth radial basis function (RBF) collo- cation methods have, under many conditions, eigenvalues with positive real part, preventing the use of such methods for time-dependent problems. We explore this difficulty at theoretical and practical levels. Theoretically, we prove that differentiation matrices for conditionally positive definite RBFs are stable for periodic domains. We also show that for Gaussian RBFs, special node distributions can achieve stability in 1-D and tensor-product nonperiodic domains. As a more practical approach for bounded domains, we consider differentiation matrices based on least-squares RBF approximations and show that such schemes can lead to stable methods on less regular nodes. By separating centers and nodes, least-squares techniques open the possibility of the separation of accuracy and stability characteristics.
Description
Keywords
radial basis functions , RBF , method of lines , numerical stability , least squares
Citation