A legendre collocation scheme for solving linear and nonlinear hyperbolic klein-gordon equations

A spectral Legendre Gauss-Lobatto collocation (L-GL-C) scheme is developed and analyzed to solve numerically nonlinear hyperbolic Klein-Gordon partial differential equations (PDEs). The method depends basically on the fact that an expansion in a series of Legendre polynomials Pnx, for the function and its space derivatives occurring in the PDE is assumed, the expansion coefficients are then determined by reducing the PDE with its boundary and initial conditions to a system of ordinary differential equations (ODEs) for these coefficients. This system may be solved analytically or numerically in step-by-step manner by using two stage implicit Runge-Kutta (IRK) method. This method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence. Numerical results indicating the high accuracy and effectiveness of this algorithm are presented. Copyright © 2015 American Scientific Publishers.