Finite element studies of the Korteweg-de Vries equation
The main aim of this study is the construction of new, efficient, and accurate numerical algorithms based on the finite element method, for the solution of the Korteweg-de Vries equation. Firstly the theoretical background to the KdV equation is discussed, and existing numerical methods based mainly on finite differences are discussed. In the following chapters finite element methods based on Bubnov-Galerkin approach are set up. Initially we used cubic Hermite interpolation functions, and in later methods cubic spline and quadratic spline shape functions. The appropriate element matrices were determined algebraically using the computer algebra package REDUCE. Finally we set up a method based on collocation using quintic spline interpolation functions. The numerical algorithms have been validated by studying the motion, interaction and deve lopment of solitons. We have demonstrated that these algorithms can faithfully represent the amplitude of a single soliton over many time steps and predict the progress of the wave front with small error. In the interaction of two solitons the numerical algorithms faithfully reproduce the changes in amplitudes and phase shifts of the analytic solution. We compare, in detai 1 the L - and L -error norms of the 2 00 present algorithms with published results. The conservative properties of the algorithms are also examined in detail. The modified and generalised Korteweg-de Vries equation have also been solved using collocation method with quintic splines interpolation functions. Again, the solution method has been validated by studying the motion, interaction, and development of solitons. We have concluded that all the new methods set up here are capable of reproducing the solutions to the KdV equation efficiently and accurately, the best amongst these methods are collocation with quintic splines or Galerkin with quadratic splines. The collocation method is also very efficient and accurate for solving the modified KdV equation.