Abstract
Differentiation matrices obtained with infinitely smooth radial basis function (RBF) collocation 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.
Original language | English (US) |
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Pages (from-to) | 1251-1268 |
Number of pages | 18 |
Journal | Computers and Mathematics with Applications |
Volume | 51 |
Issue number | 8 SPEC. ISS. |
DOIs | |
State | Published - Apr 2006 |
Externally published | Yes |
Keywords
- Least squares
- Method of lines
- Numerical stability
- RBF
- Radial basis functions
ASJC Scopus subject areas
- Modeling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics