Abstract In this paper we will show that the Runge-Kutta-Legendre (RKL) super-time-step methods are built up in stages by combining forward Euler steps with linear extrapolation steps. For second order, we will show that linear interpolation is also used. By using these characteristics a simplified algorithm will be presented. The effect of different types of external boundary conditions are shown. For Neumann (zero-flux) and Periodic the methods are shown to be monotone. For Dirichlet it is shown that there are regions of non-monotonicity where solutions have the potential to go negative. These solutions are nonphysical and will lead to erroneous results if they are feed back into system. To remove these limitations two solution strategies are presented based on different non-uniform fixed meshing philosophies. A number of applications are shown with solutions validated against analytic. For a monotone heat front and a diffused heat pulse, the RKL results are shown to be physically correct and computationally cheaper. For a compact heat pulse it will be shown that adverse effects can occur if the number of steps is too large. It will be shown that results are significantly improved by reducing the number of steps and increasing the number of outer cycles. For general applications the universal approach is to try different numbers of steps and then study any sensitivities. © British Crown Copyright AWE/2020

中文翻译：

---

Runge-Kutta-Legendre 超时间步进显式方法的一些性质

摘要 在本文中，我们将展示 Runge-Kutta-Legendre (RKL) 超时间步长方法是通过将前向欧拉步长与线性外推步长相结合来分阶段构建的。对于二阶，我们将展示也使用线性插值。通过使用这些特性，将介绍一种简化的算法。显示了不同类型的外部边界条件的影响。对于 Neumann（零通量）和 Periodic，这些方法被证明是单调的。对于狄利克雷，表明存在非单调性区域，其中解有可能变为负值。这些解决方案是非物理的，如果将它们反馈到系统中会导致错误的结果。为了消除这些限制，提出了基于不同的非均匀固定网格划分原理的两种解决方案策略。展示了许多应用程序，其中包含针对分析验证的解决方案。对于单调热锋和扩散热脉冲，RKL 结果在物理上是正确的且计算成本较低。对于紧凑的热脉冲，如果步数太大，将会出现不利影响。将表明，通过减少步骤数量和增加外部循环数量，结果得到显着改善。对于一般应用，通用方法是尝试不同数量的步骤，然后研究任何敏感性。© 英国皇冠 版权所有 AWE/2020 对于紧凑的热脉冲，如果步数太大，将会出现不利影响。将表明，通过减少步骤数量和增加外部循环数量，结果得到显着改善。对于一般应用，通用方法是尝试不同数量的步骤，然后研究任何敏感性。© 英国皇冠 版权所有 AWE/2020 对于紧凑的热脉冲，如果步数太大，将会出现不利影响。将表明，通过减少步骤数量和增加外部循环数量，结果得到显着改善。对于一般应用，通用方法是尝试不同数量的步骤，然后研究任何敏感性。© 英国皇冠 版权所有 AWE/2020