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.
The purpose of this paper is to establish a new numerical approach to solve, in two dimensions, a semilinear reaction diffusion equation combining finite volume method and Schur complement method. We applied our method for q = 2 non-overlapping subdomains and then we generalized in the case of several subdomains (q ≥ 2). A large number of numerical test cases shows the efficiency and the good accuracy of the proposed approach in terms of the CPU time and the order of the error, when increasing the number of subdomains, without using the parallel computing. After several variations of the number of subdomains and the mesh grid, we remark two significant results. On the one hand, the increase related to the number of subdomains does not affect the order of the error, on the other hand, for each mesh grid when we augment the number of subdomains, the CPU time reaches the minimum for a specific number of subdomains. In order to have the minimum CPU time, we resorted to a statistical study between the optimal number of subdomains and the mesh grid.
In this paper, we propose a vector extrapolation method for accelerating the non-overlapping Schwarz iterations in the case of the nonlinear reaction diffusion equations. The acceleration occurs at two levels: the acceleration of sequences produced by the fixed point algorithm, and the acceleration of sequences generated by Schwarz method. Specifically, the acceleration occurs in the internal Dirichlet boundary conditions. In order to illustrate the interest of the proposed algorithm, we have performed different test-cases of analytical examples, all results show the efficiency of the proposed approach in terms of CPU-Time, number of iterations and the rate of convergence.
The purpose of this work is the study of a new approach of domain decomposition method, the optimized order 4 method(OO4), to solve a reaction advection diusion equation. This method is a Schwarz waveform relaxation approach extending the known OO2 idea. The OO4 method is a reformulation of the Schwarz algorithm with specific conditions at the interface. This condition are a dierential equation of order 1 in the normal direction and of order 4 in the tangential direction to the interface resulting of artificial boundary conditions. The obtained scheme is solved by a Krylov type algorithm. The main result in this paper is that the proposed OO4 algorithm is more robust and faster than the classical OO2 method. To confirm the performance of our method , we give several numerical test-cases.
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