Convergence of an O(h 4) scheme applied to a semi-linear parabolic equation

Abstract

The quasi-linear parabolic equation u t = u xx + u yy + f(t, x, y, u) with initial boundary conditions is approximated by O(h 4) finite difference method with respect to the variable y. The resulting system of n parabolic equations is obtained in terms of the variables t and x. The maximum principle is stated and used to prove uniform convergence of the method. The system of parabolic equations is solved by the implicit finite difference method combined with Gauss–Seidel iterative method. Examples of the quasi-linear diffusion equations have been solved and the numerical results are given. The results confirm effectiveness of the methods with the small global error of the level O(h 4).