Parallel Low-Storage Runge—Kutta Solvers for ODE Systems with Limited Access Distance
Parallel Low-Storage Runge—Kutta Solvers for ODE Systems with Limited Access Distance
复制标题
用于具有有限访问距离的 ODE 系统的并行低存储 Runge — Kutta 求解器
DOI：
10.1177/1094342010384418
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
T. Rauber
中科院分区：
文献类型：
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作者：
Matthias Korch;T. Rauber
本文研究了大型常微分方程系统的初值问题的求解，其中存储空间要求决定了积分方法的选择。我们特别讨论了嵌入式龙格-库塔（RK）方法的空间高效顺序和并行实现。我们的重点是利用通常出现的ODE系统的特殊结构，称为“有限访问距离”，以提高可伸缩性和内存使用。例如，这种系统可以从偏微分方程的半离散化中产生。经典RK方法所需的存储空间与ODE系统的维数n和方法的阶数s成正比。我们提出了一种基于RK方法的流水线处理阶段的实现策略，并展示了如何通过向量的重叠将该计算方案的内存使用减少到少于三个存储寄存器，而不会影响方法系数的选择或有效步长控制的潜力。我们在不同的现代并行架构上进行了详细的运行时实验，分析和比较了不同并行实现策略的可扩展性。
We consider the solution of initial value problems (IVPs) of large systems of ordinary differential equations (ODEs) for which memory space requirements determine the choice of the integration method. In particular, we discuss the space-efficient sequential and parallel implementation of embedded Runge—Kutta (RK) methods. Our focus is on the exploitation of a special structure of commonly appearing ODE systems, referred to as ‘‘limited access distance,’’ to improve scalability and memory usage. Such systems may arise, for example, from the semi-discretization of partial differential equations (PDEs). The storage space required by classical RK methods is directly proportional to the dimension n of the ODE system and the number of stages s of the method. We propose an implementation strategy based on a pipelined processing of the stages of the RK method and show how the memory usage of this computation scheme can be reduced to less than three storage registers by an overlapping of vectors without compromising the choice of method coefficients or the potential for efficient stepsize control. We analyze and compare the scalability of different parallel implementation strategies in detailed runtime experiments on different modern parallel architectures.