Application of Petrov-Galerkin finite element method to shallow water waves model: Modified Korteweg-de Vries equation

In this article, modified Korteweg-de Vries (mKdV) equation is solved numerically by using lumped Petrov-Galerkin approach, where weight functions are quadratic and the element shape functions are cubic B-splines. The proposed numerical scheme is tested by applying four test problems including single solitary wave, interaction of two and three solitary waves, and evolution of solitons with the Gaussian initial condition. In order to show the performance of the algorithm, the error norms, L2, Loo, and a couple of conserved quantities are computed. For the linear stability analysis of numerical algorithm, Fourier method is also investigated. As a result, the computed results show that the presented numerical scheme is a successful numerical technique for solving the mKdV equation. Therefore, the presented method is preferable to some recent numerical methods.