Approximate solutions to the Gardner equation by spectral modified Exponential Time Differencing method

Abstract

This manuscript is concerned with the numerical study of spatio-temporal solution profiles of the Gardner equation by implementing a well-known spectral modified numerical scheme based on the Exponential Time Differencing method proposed by Kassam and Trefethan. The modified scheme takes full advantage of contour integral method for improving the numerical stability. But it requires judicious selection of contour path with suitable discretization to minimize the error estimates for solving a particular nonlinear partial differential equation. The efficiency of the scheme is demonstrated by testing it on some test examples – especially measuring discrete maximum norm errors and global relative errors. The associated conservative laws are also numerically verified via attaining very insignificant deviation in the conserved quantities from the initial values even after a large simulation time.