Ordinary differential equations (ODEs) are important for modelling many problems from science and engineering and efficient ODE solvers are required, for example when solving time-dependent partial differential equations (PDEs) with the method of lines. Since an ODE solver may perform a large number of iteration steps, the execution time for solving an ODE problem might be quite large. Thus, a reduction of the execution time is desirable and should affect each iteration step of the simulation. Programming techniques to reduce the execution time of ODE solver are parallelism and modification of the memory access structure such that the memory access time decreases. In this article, we investigate multithreaded solution methods for ODEs with different memory access behavior and their influence on the performance. Additionally the energy consumption is considered. The parallelism is implemented as shared memory program for multicore processors. The memory access behavior is investigated using different program variants which result from application-specific program transformations changing the memory access order while guaranteeing the numerical correctness. For the investigation of the performance, experimental data have been gathered on five different recent multicore processors. Additionally, an analytical power and energy model for modeling the performance and energy consumption is introduced. As ODE solver, the popular embedded Runge-Kutta methods with error correction is used. The simulation problems are two different ODEs resulting from discretized PDEs. The experimental data give insight into the quite diverse performance behavior of the ODE solver variants solving the same problem on different platforms.

常微分方程（ODE）对于建模科学和工程学中的许多问题非常重要，并且需要高效的ODE求解器，例如，当使用线法求解与时间有关的偏微分方程（PDE）时。由于ODE求解器可能执行大量的迭代步骤，因此解决ODE问题的执行时间可能会非常长。因此，减少执行时间是可取的，并且应该影响模拟的每个迭代步骤。减少ODE求解器执行时间的编程技术是并行性和对内存访问结构的修改，从而减少了内存访问时间。在本文中，我们研究具有不同内存访问行为的ODE的多线程解决方案方法及其对性能的影响。另外，还考虑了能耗。并行性被实现为多核处理器的共享内存程序。使用不同的程序变体来研究内存访问行为，这些变体是由特定的应用程序转换所产生的，这些更改会在保证数值正确性的同时更改内存访问顺序。为了研究性能，已经在五个不同的最新多核处理器上收集了实验数据。此外，介绍了一种用于分析性能和能耗的分析功率和能量模型。作为ODE求解器，使用了具有纠错功能的流行嵌入式Runge-Kutta方法。模拟问题是离散化的PDE导致的两个不同的ODE。