Numerical solution of quadratic general Korteweg-de Vries equation by Galerkin quadratic finite element method

摘要

In this work, we consider the quadratic generalized Kortewegde Vries (QGKdV) equation that is a mathematical model of waves on shallow water surfaces. Numerical solution of a Cauchy boundary-value problem with known exact solution is developed in details. Discretization is first accomplished by means of a quadratic finite element method. Then, the obtained system of first-order ordinary differential equations is discretized through a backward finite difference formula. Finally, the derived non linear algebraic system is solved by Newton's method with the Gauss elimination method as the inner iteration solver. Numerical results are presented in order to illustrate the efficiency of the present numerical treatment. In addition, a general form of multiple-soliton solution of QGKdV equation is obtained using the simplest equation method with Burgers equation as simplest equation.