A New Space-Time RBF Method for Reaction Diffusion and Wave Propagation Problems

This study presents a new space-time direct Kansa method for the solution of the reaction diffusion and wave propagation equations with variable coefficients. In this scheme, a new global space-time radial basis function, known as coupled space-time RBF (CST-RBF), is developed based on coupling the infinitely smooth RBF and a conic spline of appropriate order, in which a spatial and time-scale framework is introduced by treating spatial and temporal variables equally. By using the CST-RBF as the basis function, the new Kansa’s method results in a relatively well-conditioned linear system and can achieve accurate and stable numerical results without requirement of carefully selecting the shape parameter with a direct solver. Consequently, the proposed methodology completely sidesteps the bewildering issues of both ill-conditioning of the coefficient matrix and the optimal shape parameter that is a main barrier to global RBF schemes. In addition, compared with some existing space-time RBF methods, this method has higher accuracy, and can perform long-time dynamic simulation with relatively large time increments thanks to the well-conditioning of its system. Several benchmark numerical examples are well-studied to verify the accuracy and robustness of the proposed CST-RBF method.