Some essential aspects of spectral methods and their applications to PDEs

Abstract

This research study presents some essential aspects of spectral methods and their application to obtain the numerical solutions of Partial Differential Equations (PDEs). The proposed scheme is based on the spectral collocation of Lagrange and Techbychef orthogonal polynomials for both space and time integrations. The solution of PDEs is approximated as the weighted sum of the polynomials. Using collocation at the grid points, a system of equations is generated to obtain the weights of polynomials. We conclude the study by demonstrating approximate solutions for some popular PDEs. The proposed scheme achieves a greater precision with a smaller number of points than Finite Difference methods and exhibits comparatively better accuracy over Finite Element Methods in the case of approximation by higher order polynomials. This technique outperforms any existing methods of lines for time integration coupled with any scheme for spatial approximation.