A Multi-Domain Bivariate Pseudospectral Method for Evolution Equations

In this paper, we present a new general approach for solving nonlinear evolution partial differential equations. The novelty of the approach is in the combination of spectral collocation and Lagrange interpolation polynomials with Legendre-Gauss-Lobatto grid points to descritize and solve equations in piece-wise defined intervals. The method is used to solve several nonlinear evolution partial differential equations, namely, the modified KdV-Burgers equation, modified KdV equation, Fisher's equation, Burgers-Fisher equation, Burgers-Huxley equation and the Fitzhugh-Nagumo equation. The results are compared with known analytic solutions to confirm accuracy, convergence and to get a general understanding of the performance of the method. In all the numerical experiments, we report a high degree of accuracy of the numerical solutions. Strategies for implementing various boundary conditions are discussed.