Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations
Parallel-iterated Runge-Kutta methods for stiff ordinary differential equations
复制标题
刚性常微分方程的并行迭代龙格-库塔法
DOI：
10.1016/0377-0427(93)90271-c
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发表时间：
1993
期刊：
影响因子：
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通讯作者：
B. Sommeijer
中科院分区：
文献类型：
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作者：
B. Sommeijer
对于刚性常微分方程的数值积分，完全隐式龙格-库塔方法不仅具有高经典阶和高阶阶的特性，而且具有良好的稳定性。然而，这种方法需要求解一组高度耦合的阶段值方程，这是一个相当大的计算任务。本文讨论了一种解决这一问题的迭代方案。通过选择合适的迭代参数，可以将每次迭代中出现的阶段值的隐式关系解耦，从而实现并行求解。所得到的方案可以转换为对角隐式龙格-库塔（DIRK）方法类，并且与这些方法类似，每一步（每个处理器）只需要进行一次LU分解。该过程的稳定性和计算效率在很大程度上取决于迭代参数的选择和迭代次数。我们讨论了几种选择，以获得良好的稳定性和快速收敛。基于这些方法，我们编写了两个具有局部误差控制和步长变化的代码。我们在一台ALLIANT FX/4机器（四个并行向量处理器和共享内存）上实现了这两个代码，并测量了它们在许多测试问题中的加速系数。此外，还将这些代码的性能与顺序计算机上的最佳硬ODE代码（如SIMPLE、LSODE和RADAU5）的性能进行了比较。
For the numerical integration of a stiff ordinary differential equation, fully implicit Runge-Kutta methods offer nice properties, like a high classical order and high stage order as well as an excellent stability behaviour. However, such methods need the solution of a set of highly coupled equations for the stage values and this is a considerable computational task. This paper discusses an iteration scheme to tackle this problem. By means of a suitable choice of the iteration parameters, the implicit relations for the stage values, as they occur in each iteration, can be uncoupled so that they can be solved in parallel. The resulting scheme can be cast into the class of Diagonally Implicit Runge-Kutta (DIRK) methods and, similar to these methods, requires only one LU factorization per step (per processor). The stability as well as the computational efficiency of the process strongly depends on the particular choice of the iteration parameters and on the number of iterations performed. We discuss several choices to obtain good stability and fast convergence. Based on these approaches, we wrote two codes possessing local error control and stepsize variation. We have implemented both codes on an ALLIANT FX/4 machine (four parallel vector processors and shared memory) and measured their speedup factors for a number of test problems. Furthermore, the performance of these codes is compared with the performance of the best stiff ODE codes for sequential computers, like SIMPLE, LSODE and RADAU5.