Picard iteration convergence analysis in a Galerkin finite element approximation of the one‐dimensional shallow water equations
Picard iteration convergence analysis in a Galerkin finite element approximation of the one‐dimensional shallow water equations
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DOI：
10.1002/num.1690090108
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发表时间：
1993
影响因子：
3.9
通讯作者：
B. Cathers;B. O'CONNOR
中科院分区：
文献类型：
--
作者：
B. Cathers;B. O'CONNOR
本文用傅里叶方法分析了浅水模式的四种简单迭代法和Picard迭代法的收敛性。模型空间采用Galerkin线性有限元格式，时间采用隐式θ有限差分。虽然数值格式是隐式的，并且对于时间上的前向中心（θ> 0.5）是线性稳定的，但是表明如果使用Picard迭代过程，则可能存在需要观察以实现迭代收敛的附加操作限制。这些限制有效地限制了时间步长（Δt），并采用θ乘以柯朗数（θ θ）的上界的形式。在一维浅水模式中，四种迭代格式中有两种格式的迭代收敛性要求θ_（？）这些结果得到了数值实验的证实。计算机存储和运行时间之间存在权衡。运行时间最短的模型版本（由于Δt不受限制）也需要大部分计算机存储。© 1993 John Wiley & Sons，Inc.
A Fourier method for analyzing iteration convergence is applied to four different formulations of the “simple” or Picard iteration procedure of a shallow water model. The model uses the Galerkin linear FE scheme in space and the implicit θ finite differencing in time. Although the numerical scheme is implicit and linearly stable for forward centering in time (θ> 0.5), it is shown that if the Picard iteration procedure is used, there may be additional operating restrictions which need to be observed to achieve iteration convergence. These restrictions effectively limit the time step (Δt) and take the form of upper bounds on θ times the Courant number (θℂ). In the 1D shallow water model, iteration convergence requires θℂ article empty ≤1/\sqrt3 for two of the four iteration formulations, but there are no effective restrictions for the other two formulations. These results were confirmed by numerical experiments. There is a tradeoff between computer storage and run times. The model version with the shortest run time (due to unrestricted Δt) also requires most computer storage.© 1993 John Wiley & Sons, Inc.