scholarly journals The high accuracy conserved splitting domain decomposition scheme for solving the parabolic equations

2018 ◽  
Vol 3 (2) ◽  
pp. 583-592
Author(s):  
Zhongguo Zhou ◽  
Lin Li
Keyword(s):  
Domain Decomposition ◽  
Parabolic Equations ◽  
Numerical Experiments ◽  
Domain Decomposition Method ◽  
Weighted Average ◽  
Mass Conservation ◽  
High Accuracy ◽  
One Dimension ◽  
Decomposition Scheme ◽  
Time Extrapolation

AbstractIn this paper, the high accuracy mass-conserved splitting domain decomposition method for solving the parabolic equations is proposed. In our scheme, the time extrapolation and local multi-point weighted average schemes are used to approximate the interface fluxes on interfaces of sub-domains, while the interior solutions are computed by one dimension high-order implicit schemes in sub-domains. The important feature is that the developed scheme keeps mass conservation and are of second-order convergent in time and fourth-order convergent in space. Numerical experiments confirm the convergence.

A Time Second-Order Mass-Conserved Implicit-Explicit Domain Decomposition Scheme for Solving the Diffusion Equations

2017 ◽  
Vol 9 (4) ◽  
pp. 795-817 ◽  
Author(s):  
Zhongguo Zhou ◽  
Dong Liang
Keyword(s):  
Domain Decomposition ◽  
Numerical Solutions ◽  
Domain Decomposition Method ◽  
Stable Condition ◽  
Mass Conservation ◽  
Second Order ◽  
Diffusion Equations ◽  
Time And Space ◽  
Decomposition Scheme ◽  
Correction Step

AbstractIn the paper, a new time second-order mass-conserved implicit/explicit domain decomposition method (DDM) for the diffusion equations is proposed. In the scheme, firstly, we calculate the interface fluxes of sub-domains from the obtained solutions and fluxes at the previous time level, for which we apply high-order Taylor’s expansion and transfer the time derivatives to spatial derivatives to improve the accuracy. Secondly, the interior solutions and fluxes in sub-domains are computed by the implicit scheme and using the relations between solutions and fluxes, without any correction step. The mass conservation is proved and the convergence order of the numerical solutions is proved to be second-order in both time and space steps. The super-convergence of numerical fluxes is also proved to be second-order in both time and space steps. The scheme is stable under the stable conditionr≤3/5. The important feature is that the proposed domain decomposition scheme is mass-conserved and is of second order convergence in time. Numerical experiments confirm the theoretical results.

The Mass-Preserving S-DDM Scheme for Two-Dimensional Parabolic Equations

2016 ◽  
Vol 19 (2) ◽  
pp. 411-441 ◽  
Author(s):  
Zhongguo Zhou ◽  
Dong Liang
Keyword(s):  
Domain Decomposition ◽  
Parabolic Equations ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Time Step ◽  
Unconditionally Stable ◽  
One Dimensional ◽  
Splitting Technique ◽  
Theoretical Results

AbstractIn the paper, we develop and analyze a new mass-preserving splitting domain decomposition method over multiple sub-domains for solving parabolic equations. The domain is divided into non-overlapping multi-bock sub-domains. On the interfaces of sub-domains, the interface fluxes are computed by the semi-implicit (explicit) flux scheme. The solutions and fluxes in the interiors of sub-domains are computed by the splitting one-dimensional implicit solution-flux coupled scheme. The important feature is that the proposed scheme is mass conservative over multiple non-overlapping sub-domains. Analyzing the mass-preserving S-DDM scheme is difficult over non-overlapping multi-block sub-domains due to the combination of the splitting technique and the domain decomposition at each time step. We prove theoretically that our scheme satisfies conservation of mass over multi-block non-overlapping sub-domains and it is unconditionally stable. We further prove the convergence and obtain the error estimate in L2-norm. Numerical experiments confirm theoretical results.

scholarly journals Coupling Parareal with Non-Overlapping Domain Decomposition Method

2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Rim GUETAT
Keyword(s):  
Domain Decomposition ◽  
Convergence Analysis ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Domain Decomposition Method ◽  
Time Dependent ◽  
Interface Problem ◽  
Time Interval ◽  
Overlapping Domain Decomposition ◽  
Time Dependent Problems

In this paper, we present a new parallel algorithm for time dependent problems based on coupling parareal with non-overlapping domain decomposition method in order to increase parallelism in time and in space. For this we focus on the iterative methods of parallization in space to solve the interface problem like Neumann-Neumann method. In the new algorithm, the coarse temporel propagator is defined on the global domain and the Neumann-Neumann method is chosen as a fine propagator with a few iterations. We present the rigorous convergence analysis of the new coupled algorithm on bounded time interval. Numerical experiments illustrate the performance of this new algorithm and confirm our analysis. RÉSUMÉ. Dans ce papier, nous présentons un nouvel algorithme parallèle pour les problèmes dé-pendant du temps basé sur le couplage du pararéel avec les méthodes de décomposition de domaine sans recouvrement afin d'augmenter le parallélisme dans le temps et l'espace. Nous nous concen-trons sur les méthodes itératives de parallélisation en espace pour résoudre le problème d'interface par la méthode de Neumann-Neumann. Dans ce nouvel algorithme, le propagateur grossier est dé-finie sur le domaine global et la méthode de Neumann-Neumann est choisi pour le propagateur fin avec quelques itérations. Nous présentons l'analyse rigoureuse de convergence du nouvel algorithme couplé sur un intervalle de temps borné. Des expèriences numériques illustrent les performances de ce nouvel algorithme et confirment notre analyse. Dans ce papier, nous présentons un nouvel algorithme parallèle pour les problèmes dépendantdu temps basé sur le couplage du pararéel avec les méthodes de décomposition de domainesans recouvrement afin d’augmenter le parallélisme dans le temps et l’espace. Nous nous concentronssur les méthodes itératives de parallélisation en espace pour résoudre le problème d’interfacepar la méthode de Neumann-Neumann. Dans ce nouvel algorithme, le propagateur grossier est définiesur le domaine global et la méthode de Neumann-Neumann est choisi pour le propagateur finavec quelques itérations. Nous présentons l’analyse rigoureuse de convergence du nouvel algorithmecouplé sur un intervalle de temps borné. Des expèriences numériques illustrent les performances dece nouvel algorithme et confirment notre analyse.

scholarly journals PREDICTOR-CORRECTOR DOMAIN DECOMPOSITION ALGORITHM FOR PARABOLIC PROBLEMS ON GRAPHS

2012 ◽  
Vol 17 (1) ◽  
pp. 113-127 ◽  
Author(s):  
Natalija Tumanova
Keyword(s):  
Domain Decomposition ◽  
Convergence Analysis ◽  
Domain Decomposition Method ◽  
Energy Estimates ◽  
Parabolic Problems ◽  
Decomposition Scheme ◽  
Predictor Corrector ◽  
Unsteady Problems ◽  
Domain Decomposition Algorithm ◽  
Predictor Step

In this paper, we present a predictor-corrector type algorithm for solution of linear parabolic problems on graph structure. The graph decomposition is done by dividing some edges and therefore we get a set of problems on sub-graphs, which can be solved efficiently in parallel. The convergence analysis is done by using the energy estimates. It is proved that the proposed finite-difference scheme is unconditionally stable but the predictor step error gives only conditional approximation. In the second part of the paper it is shown that the presented algorithm can be written as Douglas type scheme, based on the domain decomposition method. For a simple case of one dimensional parabolic problem, the convergence analysis is done by using results from [P. Vabishchevich. A substracturing domain decomposition scheme for unsteady problems. Comp. Meth. Appl. Math. 11(2):241-268, 2011]. The optimality of asymptotical error estimates is investigated. Results of computational experiments are presented.

scholarly journals Hybrid Discontinuous Galerkin Discretisation and Domain Decomposition Preconditioners for the Stokes Problem

2019 ◽  
Vol 19 (4) ◽  
pp. 703-722 ◽  
Author(s):  
Gabriel R. Barrenechea ◽  
Michał Bosy ◽  
Victorita Dolean ◽  
Frédéric Nataf ◽  
Pierre-Henri Tournier
Keyword(s):  
Domain Decomposition ◽  
Discontinuous Galerkin ◽  
Numerical Experiments ◽  
Decomposition Method ◽  
Degrees Of Freedom ◽  
Domain Decomposition Method ◽  
Stokes Problem ◽  
Stability And Convergence ◽  
Interface Conditions ◽  
Optimal Domain

AbstractSolving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of nonstandard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the nonstandard interface conditions are naturally defined at the boundary between elements. In this paper, we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with nonstandard boundary conditions. The full stability and convergence analysis of the discretisation method is presented, and the results are corroborated by numerical experiments. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

Sign in / Sign up

Export Citation Format

Share Document