Spectral Analysis on Burgers’ Equation and Its Solutions Using Three Different Basis Functions

Abstract

In this study, Spectral Collocation method with three different basis functions are used to analyse the famous Burgers’ equation. It is the simplest nonlinear model which represents diffusive waves in fluid dynamics and is a quasi-linear parabolic partial differential equation which is a balance between nonlinearity, time evolution and diffusion. Though several methods of solution procedures are available, it was found that for a very small viscosity, the capture of shock waves was not well observed by most of the work present in the literature. With the present analysis, we show that with few number of collocation points i.e. with less computational effort, it was possible to capture the shock formations. The three basis functions considered in our analysis are the Chebyshev Polynomials, the Legendre Polynomials and the Jacobi polynomials. Their role in obtaining more accurate solutions is the highlighted in this work. Several available results in the latest literature are compared to show the efficiency of the present method of solution. We have considered seven different cases of initial and boundary conditions. For three cases for which already analytical solutions are available in the literature, our methods of solutions show a very good agreement. Two sets of new initial and boundary conditions have been considered by us and their results are computed. It can be observed that for all the computations, the number of collocation points required were only few to get accurate solutions. Several tables and figures are presented to support our analysis.