Numerical solutions of nonlinear evolutionary problems : Generalized finite difference methods and linear implicit methods

Abstract

In this thesis new methods are developed for two nonlinear wave equations: Burgers equation and the KdV equation. Generalized finite difference schemes are used to approximate the space variables and linear implicit methods for the time stepping. Many numerical experiments have been carried out to verify and support this work. Generalized finite difference methods, which achieve higher order space discretization without using additional nodal points, give more accurate numerical results with little additional computational cost because the difference of computational costs between solving an implicit ODE and an explicit ODE is little if implicit time stepping is used. The new 4-stage order 3 linear implicit method (the W-method) gives quite promising numerical results for both short- and long-term time integration. The new method is more efficient than the standard Crank-Nicolson method. The new method has appropriate stability properties and its implementation is simpler than implicit Runge-Kutta methods. Important observations on the W-method are that the parameter γ affects the performance of the method, and should be chosen so that it is .Lp-stable; and that the parameter j should be chosen as a good approximation to the Jacobian matrix. The W-method was observed to be stable for the KdV equation if ║J – J║=O(τ). The Crank-Nicolson method is a safe method but its computational costs are almost twice as much as for the new method W3B. The long term time integration performances of the 4-stage order 4 Rosenbrock method and the 2-stage order 2 W-methods are very disappointing. By comparison, W3B seems to be a promising alternative for long term time integration, as it is fast and can produce accurate results.