Adaptive radial basis function for time dependent partial differential equations

We propose a meshless adaptive solution of the time-dependent partial differential equations (PDE) using radial basis functions (RBFs). The approximate solution to the PDE is obtained using multiquadrics (MQ). We choose MQ because of its exponential convergence for sufficiently smooth functions. The solution of partial differential equations arising in science and engineering frequently have large variations occurring over small portion of the physical domain. The challenge then is to resolve the solution behaviour there. For the sake of efficiency we require a finer grid in those parts of the physical domain whereas a much coarser grid can be used otherwise. Local scattered data reconstruction is used to compute an error indicator to decide where nodes should be placed. We use polyharmonic spline approximation in this step. The performance of the method is shown for numerical examples of one dimensional Kortwegde-Vries equation, Burger’s equation and Allen-Cahn equation.