Monotone Iterative Numerical Method For A Class Of Nonlinear Diffusion Equations

This paper mainly studies the numerical solution of a class of nonlinear diffusion equations using a monotone iteration method with the global convergence . The equation we study plays a fundamental role in the modelling of various processes of heat conduction, reaction-diffusions, in mathematical biology, and in many other fields.We construct a appropriate iteration scheme to compute the discrete system of this equation and we present a new existence condition and constructive method of the initial iteration values. Finally, we combine the monotone iteration method with the parallel algorithm.In Chapter 1, we introduce the physics background of this class of nonlinear diffusion equation and generalize the monotone iteration.In Chapter 2, we mainly present a new existence condition and constructive method of the initial iteration values, and construct a appropriate iteration scheme to computer the discrete system of this equation. First, we find a suitable difference method that can ensure the absolute stability of the diffusion scheme. Also the difference method makes sure that we can use the monotone iteration to compute the discrete equations that brought by the diffusion scheme. Second, we present a new existence condition and a constructive method of the initial iteration values . Third, we construct a monotone iteration scheme according to the characteristic of the discrete equations.In Chapter 3, we prove the convergence of the monotone iteration scheme and the theorem of unique solution. If we use the upper and lower solution as the initial iteration values and compute using the monotone iteration scheme, we can prove that the two iteration sequences converge. Also we prove that the solution of the discrete equations is unique when the time increment satisfies a certain conduction.In Chapter 4, we introduce a parallel algorithm and combine the monotone iteration method with the parallel algorithm to solve the discrete equations. The efficiency of the computation is improved.In Chapter 5, a numerical example is presented. We use the method in this paper to compute the numerical solution of a initial and edge value problem of a Burgers-Huxley equation. The result shows the efficiency of our method.