Two results on polynomial interpolation in equally spaced points
Journal of Approximation Theory
A modified Chebyshev pseudospectral method with an O(N–1) time step restriction
Journal of Computational Physics
An efficient numerical scheme for Burgers' equation
Applied Mathematics and Computation
Spectral methods in MatLab
Accuracy, Resolution, and Stability Properties of a Modified Chebyshev Method
SIAM Journal on Scientific Computing
Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation
SIAM Journal on Numerical Analysis
Computing eigenmodes ofelliptic operators using radial basis functions
Computers & Mathematics with Applications
On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere
Journal of Computational Physics
Stabilization of RBF-generated finite difference methods for convective PDEs
Journal of Computational Physics
Journal of Scientific Computing
Vector field approximation using radial basis functions
Journal of Computational and Applied Mathematics
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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.