Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation
Space-Time Continuous Analysis of Waveform Relaxation for the Heat Equation
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DOI：
10.1137/s1064827596305337
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发表时间：
1998-11
期刊：
影响因子：
--
通讯作者：
M. Gander;A. Stuart
中科院分区：
文献类型：
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作者：
M. Gander;A. Stuart
偏微分方程的波形松弛算法传统上是通过将偏微分方程在空间离散化，然后使用矩阵分裂来分裂离散算子。对于半离散的热传导方程，该方法可以证明在无界时间区间上的线性收敛性和在有界时间区间上的超线性收敛性。然而，边界一般取决于网格参数和收敛速度恶化，因为一个细化的网格。在电路仿真中，波形松弛的原始发展的动机，其中的电路被分裂成子电路的物理域中，我们分裂的PDE使用重叠域分解。我们证明了线性收敛的算法在连续的情况下，在一个无限的时间间隔，根据重叠的大小的速度。这一结果在空间离散化后仍然有效，收敛速度相对于网格细化是鲁棒的。该算法属于基于重叠多分裂的波形松弛算法类。我们的分析量化了Jeltsch和Pohl [SIAM J. Sci.计算：第16（1995）号来文，第16页。40--49]，多分裂算法的收敛速度取决于重叠。数值结果支持收敛理论。
Waveform relaxation algorithms for partial differential equations (PDEs) are traditionally obtained by discretizing the PDE in space and then splitting the discrete operator using matrix splittings. For the semidiscrete heat equation one can show linear convergence on unbounded time intervals and superlinear convergence on bounded time intervals by this approach. However, the bounds depend in general on the mesh parameter and convergence rates deteriorate as one refines the mesh. Motivated by the original development of waveform relaxation in circuit simulation, where the circuits are split in the physical domain into subcircuits, we split the PDE by using overlapping domain decomposition. We prove linear convergence of the algorithm in the continuous case on an infinite time interval, at a rate depending on the size of the overlap. This result remains valid after discretization in space and the convergence rates are robust with respect to mesh refinement. The algorithm is in the class of waveform relaxation algorithms based on overlapping multisplittings. Our analysis quantifies the empirical observation by Jeltsch and Pohl [SIAM J. Sci. Comput., 16 (1995), pp. 40--49] that the convergence rate of a multisplitting algorithm depends on the overlap. Numerical results are presented which support the convergence theory.