Spectral methods using Legendre wavelets for nonlinear Klein∖∖Sine-Gordon equations

Klein/Sine-Gordon equations are very important in that they can accurately model many essential physical phenomena. In this paper, we propose a new spectral method using Legendre wavelets as basis for numerical solution of Klein∖∖Sine-Gordon Equations. Due to the good properties of wavelets basis, the proposed method can obtain good spatial and spectral resolution. Moreover, the presented method can save more memory and computation time benefit from save more computation time benefit from the hierarchical scale structure of Legendre wavelets. 1D and 2D examples are included to demonstrate the validity and applicability of the new technique. Numerical results show the exponential convergence property and error characteristics of presented method.