Abstract
This paper concerns with the analysis of the iterative procedure for the solution of a nonlinear reaction diffusion equation at the steady state in a two dimensional bounded domain supplemented by suitable boundary conditions. This procedure, called Lagged Diffusivity Functional Iteration (LDFI)-procedure, computes the solution by “lagging” the diffusion term. A model problem is considered and a finite difference discretization for that model problem is described. Furthermore, properties of the finite difference operator are proved. Then, sufficient conditions for the convergence of the LDFI-procedure are given. At each stage of the LDFI-procedure a weakly nonlinear algebraic system has to be solved and the simplified Newton–Arithmetic Mean (Newton–AM) method is used. This method is particularly well suited for implementation on parallel computers. Numerical studies show the efficiency, for different test functions, of the LDFI-procedure combined with the simplified Newton–AM method. Better results are obtained when in the reaction diffusion equation also a convection term is present.
Highlights
We consider a nonlinear steady state reaction diffusion equation where the diffusion coefficient and the rate of change due to a reaction depend on the solution
In this paper we have analysed the Lagged Diffusivity Functional Iteration (LDFI)–procedure combined with the simplified Newton–Arithmetic Mean (AM) method for the solution of finite difference nonlinear systems
2 and 3, we observe that when the values of the function g(φ) are rapidly increasing for 0 ≤ φ ≤ 1 or the values of the function α(x, y) are large, the diagonal of the matrix Cν becomes more dominant; it implies a reduction of the total number of the simplified Newton iterations;
Summary
We consider a nonlinear steady state reaction diffusion equation where the diffusion coefficient and the rate of change due to a reaction depend on the solution. A purpose here is to re-examine the LDFI–procedure for solving the system of nonlinear difference equations of elliptic type in the context of Parallel Computing, a simplified version of the Newton–Arithmetic Mean (Newton–AM) method ([9]) is the inner iterative solver used for the solution of the weakly nonlinear systems. This is a two–stage iterative method and is suited for implementation on parallel computers ([7], [8]; see [1], [2], [3], [32]).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Communications in Nonlinear Science and Numerical Simulation
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
