New optimized domain decomposition order 4 method(OO4) applied to reaction advection diffusion equation

2018 ◽  
Vol 9 (1-2) ◽  
pp. 28-41
Author(s):  
M. R. Amattouch ◽  
H. Belhadj ◽  
N. Nagid
Keyword(s):  
Domain Decomposition ◽  
Domain Decomposition Method ◽  
Numerical Test ◽  
Tangential Direction ◽  
Waveform Relaxation ◽  
Test Cases ◽  
New Approach ◽  
Artificial Boundary Conditions ◽  
Advection Diffusion Equation ◽  
Schwarz Algorithm

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.

scholarly journals Optimized Domain Decomposition Method for Non Linear Reaction Advection Diffusion Equation

2016 ◽  
Vol 12 (27) ◽  
pp. 63 ◽  
Author(s):  
M.R. Amattouch ◽  
N. Nagid ◽  
H. Belhadj
Keyword(s):  
Diffusion Equation ◽  
Domain Decomposition ◽  
Rate Of Convergence ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Linear Reaction ◽  
Non Linear ◽  
Schwarz Algorithm

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.

ON THE ELLIPTIC-HYPERBOLIC COUPLING I: THE ADVECTION-DIFFUSION EQUATION VIA THE χ-FORMULATION

1993 ◽  
Vol 03 (02) ◽  
pp. 145-170 ◽  
Author(s):  
C. CANUTO ◽  
A. RUSSO
Keyword(s):  
Diffusion Equation ◽  
Domain Decomposition ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Advection Equation ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Subdomain Method ◽  
Self Adaptive

An advection-diffusion equation is considered, for which the solution is advection-dominated in most of the domain. A domain decomposition method based on a self-adaptive, smooth coupling of the reduced advection equation and the full advection-diffusion equation is proposed. The convergence of an iteration-by-subdomain method is investigated.

scholarly journals Schwarz Waveform Relaxation for Heat Equations with Nonlinear Dynamical Boundary Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Shu-Lin Wu
Keyword(s):  
Boundary Conditions ◽  
Domain Decomposition ◽  
Domain Decomposition Method ◽  
Linear Convergence ◽  
Decomposition Methods ◽  
Waveform Relaxation ◽  
Heat Equations ◽  
Nonlinear Dynamical ◽  
Schwarz Waveform Relaxation ◽  
Dynamical Boundary Conditions

We are interested in solving heat equations with nonlinear dynamical boundary conditions by using domain decomposition methods. In the classical framework, one first discretizes the time direction and then solves a sequence of state steady problems by the domain decomposition method. In this paper, we consider the heat equations at spacetime continuous level and study a Schwarz waveform relaxation algorithm for parallel computation purpose. We prove the linear convergence of the algorithm on long time intervals and show how the convergence rate depends on the size of overlap and the nonlinearity of the nonlinear boundary functions. Numerical experiments are presented to verify our theoretical conclusions.

Spectral‐domain decomposition methods for the solution of acoustic and elastic wave equations

Geophysics ◽  
1996 ◽  
Vol 61 (4) ◽  
pp. 1160-1174 ◽  
Author(s):  
Ezio Faccioli ◽  
Fabio Maggio ◽  
Alfio Quarteroni ◽  
Aldo Taghan
Keyword(s):  
Domain Decomposition ◽  
Wave Equations ◽  
Heterogeneous Media ◽  
Domain Decomposition Method ◽  
Decomposition Methods ◽  
Spectral Domain ◽  
Test Cases ◽  
Domain Decomposition Methods ◽  
Elastic Wave Equations ◽  
The Stability

A new spectral‐domain decomposition method is presented for acoustic and elastodynamic wave propagation in 2-D heterogeneous media. Starting from a variational formulation of the problem, two different approaches are proposed for the spatial discretization: a mixed Fourier‐Legendre and a full Legendre collocation. The matching conditions at subdomain interfaces are carefully analyzed, and the stability and efficiency of time‐advancing schemes are investigated. The numerical validation with some significant test cases illustrates the accuracy, flexibility, and robustness of our methods. These allow the treatment of complex geometries and heterogeneous media while retaining spectral accuracy.

scholarly journals A non-overlapping Schwarz domain decomposition method with high-order finite elements for flow acoustics

2020 ◽  
Vol 369 ◽  
pp. 113223
Author(s):  
Alice Lieu ◽  
Philippe Marchner ◽  
Gwénaël Gabard ◽  
Hadrien Bériot ◽  
Xavier Antoine ◽  
...  
Keyword(s):  
Finite Elements ◽  
Domain Decomposition ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
High Order ◽  
Overlapping Schwarz
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