2,348 research outputs found
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
Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation for the Wave Equation
We present a Waveform Relaxation (WR) version of the Dirichlet-Neumann and
Neumann-Neumann algorithms for the wave equation in space time. Each method is
based on a non-overlapping spatial domain decomposition, and the iteration
involves subdomain solves in space time with corresponding interface condition,
followed by a correction step. Using a Laplace transform argument, for a
particular relaxation parameter, we prove convergence of both algorithms in a
finite number of steps for finite time intervals. The number of steps depends
on the size of the subdomains and the time window length on which the
algorithms are employed. We illustrate the performance of the algorithms with
numerical results, and also show a comparison with classical and optimized
Schwarz WR methods.Comment: 8 pages, 6 figures, presented in 22nd International conference on
Domain Decomposition Methods, to appear in Domain Decomposition in Science
and Engineering XXII, LNCSE, Springer-Verlag 201
A fast solver for systems of reaction-diffusion equations
In this paper we present a fast algorithm for the numerical solution of
systems of reaction-diffusion equations, , , . Here,
is a vector-valued function, , is
large, and the corresponding system of ODEs, , is
stiff. Typical examples arise in air pollution studies, where is the given
wind field and the nonlinear function models the atmospheric chemistry.Comment: 8 pages, 3 figures, to appear in Proc. 13th Domain Decomposition
Conference, Lyon, October 200
Pipeline Implementations of Neumann-Neumann and Dirichlet-Neumann Waveform Relaxation Methods
This paper is concerned with the reformulation of Neumann-Neumann Waveform
Relaxation (NNWR) methods and Dirichlet-Neumann Waveform Relaxation (DNWR)
methods, a family of parallel space-time approaches to solving time-dependent
PDEs. By changing the order of the operations, pipeline-parallel computation of
the waveform iterates are possible without changing the final solution. The
parallel efficiency and the increased communication cost of the pipeline
implementation is presented, along with weak scaling studies to show the
effectiveness of the pipeline NNWR and DNWR algorithms.Comment: 20 pages, 8 figure
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