28,376 research outputs found
An asynchronous three-field domain decomposition method for first-order evolution problems
summary:We present an asynchronous multi-domain time integration algorithm with a dual domain decomposition method for the initial boundary-value problems for a parabolic equation. For efficient parallel computing, we apply the three-field domain decomposition method with local Lagrange multipliers to ensure the continuity of the primary unknowns at the interface between subdomains. The implicit method for time discretization and the multi-domain spatial decomposition enable us to use different time steps (subcycling) on different parts of a computational domain, and thus efficiently capture the underlying physics with less computational effort. We illustrate the performance of the proposed multi-domain time integrator by means of a simple numerical example
A combined fractional step domain decomposition method for the numerical integration of parabolic problems
In this paper we develop parallel numerical algorithms to
solve linear time dependent coefficient parabolic problems. Such methods
are obtained by means of two consecutive discretization procedures.
Firstly, we realize a time integration of the original problem using a
Fractional Step Runge Kutta method which provides a family of elliptic
boundary value problems on certain subdomains of the original domain.
Next, we discretize those elliptic problems by means of standard techniques.
Using this framework, the numerical solution is obtained by solving,
at each stage, a set of uncoupled linear systems of low dimension.
Comparing these algorithms with the classical domain decomposition
methods for parabolic problems, we obtain a reduction of computational
cost because of, in this case, no Schwarz iterations are required. We give
an unconditional convergence result for the totally discrete scheme and
we include two numerical examples that show the behaviour of the proposed
method.This research is partially supported by the MCYT research project num. BFM2000-0803 and the research project resolution 134/2002 of Government of Navarra
A Time-Dependent Dirichlet-Neumann Method for the Heat Equation
We present a waveform relaxation version of the Dirichlet-Neumann method for
parabolic problem. Like the Dirichlet-Neumann method for steady problems, the
method is based on a non-overlapping spatial domain decomposition, and the
iteration involves subdomain solves with Dirichlet boundary conditions followed
by subdomain solves with Neumann boundary conditions. However, each subdomain
problem is now in space and time, and the interface conditions are also
time-dependent. Using a Laplace transform argument, we show for the heat
equation that when we consider finite time intervals, the Dirichlet-Neumann
method converges, similar to the case of Schwarz waveform relaxation
algorithms. The convergence rate depends on the length of the subdomains as
well as the size of the time window. In this discussion, we only stick to the
linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and
Engineering, Vol. 98, Springer-Verlag 201
The PDD method for solving linear, nonlinear, and fractional PDEs problems
We review the Probabilistic Domain Decomposition (PDD) method for the numerical solution of linear and nonlinear Partial Differential Equation (PDE) problems. This Domain Decomposition (DD) method is based on a suitable probabilistic representation of the solution given in the form of an expectation which, in turns, involves the solution of a Stochastic Differential Equation (SDE). While the structure of the SDE depends only upon the corresponding PDE, the expectation also depends upon the boundary data of the problem. The method consists of three stages: (i) only few values of the sought solution are solved by Monte Carlo or Quasi-Monte Carlo at some interfaces; (ii) a continuous approximation of the solution over these interfaces is obtained via interpolation; and (iii) prescribing the previous (partial) solutions as additional Dirichlet boundary conditions, a fully decoupled set of sub-problems is finally solved in parallel. For linear parabolic problems, this is based on the celebrated Feynman-Kac formula, while for semilinear parabolic equations requires a suitable generalization based on branching diffusion processes. In case of semilinear transport equations and the Vlasov-Poisson system, a generalization of the probabilistic representation was also obtained in terms of the Method of Characteristics (characteristic curves). Finally, we present the latest progress towards the extension of the PDD method for nonlocal fractional operators. The algorithm notably improves the scalability of classical algorithms and is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Numerical examples conducted in 1D and 2D, including some for the KPP equation and Plasma Physics, are given.info:eu-repo/semantics/acceptedVersio
Stochastic domain decomposition for time dependent adaptive mesh generation
The efficient generation of meshes is an important component in the numerical solution of problems in physics and engineering. Of interest are situations where global mesh quality and a tight coupling to the solution of the physical partial differential equation (PDE) is important. We consider parabolic PDE mesh generation and present a method for the construction of adaptive meshes in two spatial dimensions using stochastic domain decomposition that is suitable for an implementation in a multi– or many–core environment. Methods for mesh generation on periodic domains are also provided. The mesh generator is coupled to a time dependent physical PDE and the system is evolved using an alternating solution procedure. The method uses the stochastic representation of the exact solution of a parabolic linear mesh generator to find the location of an adaptive mesh along the (artificial) subdomain interfaces. The deterministic evaluation of the mesh over each subdomain can then be obtained completely independently using the probabilistically computed solutions as boundary conditions. The parallel performance of this general stochastic domain decomposition approach has previously been shown. We demonstrate the approach numerically for the mesh generation context and compare the mesh obtained with the corresponding single domain mesh using a representative mesh quality measure
A Class of Stable, Globally Noniterative, Nonoverlapping Domain Decomposition Algorithms for the Simulation of Parabolic Evolutionary Systems.
Parabolic systems are governed by time dependent partial differential equations. To obtain a high simulation quality that captures important features of a parabolic system requires solving the governing equation to an adequate accuracy, which necessitates a large sampling size in the spatial and temporal dimensions, and hence a large amount of simulation data and high computing cost. Domain decomposition is an effective method of divide-and-conquer paradigm that divides the problem domain into several subdomains, reducing the original problem into several smaller interdependent problems which can be solved in parallel. In this dissertation, we propose a class of stabilized explicit-implicit time marching (SEITM) domain decomposition algorithms for parabolic equations. Explicit-implicit time marching (EITM) algorithms are globally non-iterative nonoverlapping domain decomposition methods, which, when compared with Schwartz algorithm based parabolic solvers, are both computationally and communicationally efficient for each time step simulation but suffer from small time step size restrictions due to conditional stability. The proposed stabilization techniques in the SEITM algorithms retain the time-stepwise efficiency in computation and communication of the EITM algorithms but free the algorithms from small time step size restrictions, rendering SEITM algorithms excellent candidates for large scale parallel simulation problems. Three algorithms of the SEITM class are presented in this dissertation, which are mathematically analyzed and experimentally tested to show excellent numerical stability, computation and communication efficiencies, and high parallel speedup and scalability
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