Application of Petrov-Galerkin Finite Element Method to Shallow Water Waves Model: Modified Korteweg-de Vries Equation

In this article a Petrov-Galerkin method in which the element shape functions are cubic and weight functions are quadratic B-splines, is introduced to solve the modified Korteweg-de Vries (mKdV) equation. Solitary wave motion, interaction of two and three solitary waves and the development of the Gaussian initial condition into solitary waves are studied using the proposed method. Accuracy and efficiency of the method are demonstrated by computing the numerical conserved laws and $L_{2}$, $L_{infty }$ error norms. The computed results show that the present scheme is a successful numerical technique for solving the mKdV equation. A linear stability analysis based on the Fourier method is also investigated.