NUMERICAL SOLUTION OF SOME NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS USING VARIABLE MESH TECHNIQUE

Abstract

Implicit methods with their superior stability properties, are almost used to solve non-uniform partial differential equations. We present implicit techniques with nonlinear spatial step to find numerical solutions of the nonlinear BBM equation and Burgers' equation. The nonlinearity in the resulting implicit schemes is avoided by using Picard linearization method. At each time level, tridiagonal system of linear equations of the Burgers' equation with time independent matrix of coefficient is solved by using Thomas algorithm (ch.II) to avoid round-off error growth and minimize the storage in the machine computation, and five diagonal system of linear equations of the BBM equation with time independent matrix of coefficient is solved as shown in ch.III. The interaction of the solitary waves for the BBM equation (ch.III) is investigated using implicit scheme with variable mesh, i.e., a grid where the mesh can be refined or enlarged following the behavior of the solution. Burgers' equation (ch.IV) is investigated (in two different initial and boundary conditions) using implicit scheme with non-uniform spatial step as in the BBM equation. The method presented in this thesis is very useful for some problems (if the solution is steep in some region and smooth in the other). In these problems, the mesh is refined in the region of steep gradients and enlarged in smoother regions. In this method, we minimized the number of points to be calculated and minimize the time of the computation to calculate the extra points. In the same time, we have almost the same error if we use constant fine mesh instead of the variable mesh.