A fast algorithm for particle simulations
Journal of Computational Physics
Boundary Layer Resolving Pseudospectral Methods for Singular Perturbation Problems
SIAM Journal on Scientific Computing
A new numerical algorithm for the analytic continuation of Green's functions
Journal of Computational Physics
A Fast Adaptive Numerical Method for Stiff Two-Point Boundary Value Problems
SIAM Journal on Scientific Computing
A fast spectral algorithm for nonlinear wave equations with linear dispersion
Journal of Computational Physics
Multiquadric collocation method with integralformulation for boundary layer problems
Computers & Mathematics with Applications
Numerical simulations of 2D fractional subdiffusion problems
Journal of Computational Physics
An adaptive least-squares collocation radial basis function method for the HJB equation
Journal of Global Optimization
Hi-index | 7.30 |
An adaptive collocation method based upon radial basis functions is presented for the solution of singularly perturbed two-point boundary value problems. Using a multiquadric integral formulation, the second derivative of the solution is approximated by multiquadric radial basis functions. This approach is combined with a coordinate stretching technique. The required variable transformation is accomplished by a conformal mapping, an iterated sine-transformation. A new error indicator function accurately captures the regions of the interval with insufficient resolution. This indicator is used to adaptively add data centres and collocation points. The method resolves extremely thin layers accurately with fairly few basis functions. The proposed adaptive scheme is very robust, and reaches high accuracy even when parameters in our coordinate stretching technique are not chosen optimally. The effectiveness of our new method is demonstrated on two examples with boundary layers, and one example featuring an interior layer. It is shown in detail how the adaptive method refines the resolution.