Numerical methods for the solution of the third- and fifth-order dispersive Korteweg-de Vries equations

Two numerical methods are proposed for the solution of the third- and fifth-order Korteweg-de Vries equations. The first method is derived using central differences to replace the space derivatives with a predictor-corrector time-stepping and the second method by linearizing the implicit corrector scheme in which the solution is then found by solving a linear algebraic system at each time step rather than a nonlinear algebraic system which is more usual. ortant advantage to be gained from the use of the linearized implicit method over the predictor-corrector method which is optimally stable, is the ability to vary the mesh length. thods are analysed with respect to stability criteria and numerical dispersion. Numerical results portraying a single soliton solution and the interaction of more than one soliton are reported for the third-order Korteweg-de Vries equation. Numerical results for the fifth-order Korteweg-de Vries equation using the linearized implicit method are also reported.