Optimized Domain Decomposition Method for Non Linear Reaction Advection Diffusion Equation

Abstract

This work is devoted to an optimized domain decomposition method applied to a non linear reaction advection diffusion equation. The proposed method is based on the idea of the optimized of two order (OO2) method developed this last two decades. We first treat a modified fixed point technique to linearize the problem and then we generalize the OO2 method and modify it to obtain a new more optimized rate of convergence of the Schwarz algorithm. To compute the new rate of convergence we have used Fourier analysis. For the numerical computation we minimize this rate of convergence using a global optimization algorithm. Several test-cases of analytical problems illustrate this approach and show the efficiency of the proposed new method.

A modified fixed point method for biochemical transport

“…) is equivalent to the global minimization of the Energy E on the space H with some constraints. We can prove that E is a K contraction on V for a small number M (see [1], [2] for prove and explanation in the case of nonlinear diffusion). This method is local and could diverge for an initial sequence departure is u 0 = h 0 .…”

“…As a solution to this problem we make some modifications to the equation to have successive sequences that are not distant from the initial value u 0 = h 0 . As in [2] where, we proposed a modified fixed point to a just a semi linear equation, and in [1] where, we applied a modified fixed point to a nonlinear equation, we generalize this method to our equation by mean of solving the following iterative equations:…”

“…Also, the use of explicit scheme to solve these equations is not accurate, stable and efficient in general . So In order to make ours bio-chemicals simulations fast, stable and efficient, we apply in this paper, the modified fixed point (we propose in [1], [2], [4]) to solve problem (1.1) type.…”

A Modified Fixed Point Method for the Perona Malik Equation

“…Domain decomposition methods can reduce this cost by splitting initial problem into two or more sub-problems with smaller dimensions. Many authors have studied domain decomposition methods these last decades [17,18]. Among these methods we consider in this work the method called second order optimized method OO2.…”