Finite element studies of the modified KdV equation
The main aim of this study is the construction of new efficient and accurate numerical algorithms based on the B-spline finite element method, for solution of the Korteweg-de Vries (KdV) and Modified Korteweg-de Vries (MKdV) equations. In the following chapters; the theoretical background to the KdV and MKdV equations is discussed, and existing numerical methods are described. Numerical solutions to the KdV and MKdV equations are obtained using the Galerkin and modified Petrov-Galerkin method with quadratic B-spline finite elements over which the non-linear term is locally linearised. The numerical algorithms have been validated by studying the motion, interaction and development of solitons. We have demonstrated that these algorithms can faithfully represent the amplitude of a single soliton over many time steps and the interaction of two solitons. A new numerical solution for the MKdV - equation is obtained using a "lumped" Galerkin method with quadratic 13- spline finite elements. The motion, interaction and generation of solitary waves are studied using the method. An unconditionally stable numerical algorithm is implemented for the solution of the MKdV equation using a collocation method with quartic 13- spline finite elements. The algorithm is validated through a single soliton simulation. In further numerical experiments forced boundary conditions u= Uo are applied at the end x=0 and the generated states of solitary waves are studied. The solitary wave states generated by applying a positive impulse followed immediately by an equal negative impulse is dependent on the period of forcing. The solitary waves generated by these various forcing functions possess many of the attributes of free solitons.