Eigenvalue stability of radial basis function discretizations for time-dependent problems
Date
2005
Authors
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