论文标题
具有错误估计器
Very High-Order A-stable Stiffly Accurate Diagonally Implicit Runge-Kutta Methods with Error Estimators
论文作者
论文摘要
一种数值搜索方法用于设计配备嵌入式误差估计器的高阶对角线runge-kutta（Dirk）方案，其中一些具有相同的对角线元素（SDIRK）和显式第一阶段（ESDIRK）。在这些类中的每个类别中，我们提出了六个订单的新的A稳定方案（先前已知的A稳定性Dirk型方案的最高顺序），直到八订单。对于每个顺序，我们都包括一个仅是稳定的方案以及L稳定，僵硬和/或具有阶段阶阶的方案。后一种类型需要更多的阶段，但对于差分 - 地球方程（DAE）提供了更好的收敛速率，并且具有阶段订单二的阶段的收敛速率可以更好地适应中等严格的问题。除了实施A稳定性外，还需要为超过100个变量的200个方程式找到高度准确的数值解决方案，而这是通过全球和局部优化策略的结合来完成的，这还需要为200个方程的系统找到高度准确的数值解决方案。在各种问题上证明了对方案的准确性，稳定性和适应性步骤的控制。
A numerical search approach is used to design high-order diagonally implicit Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of which have identical diagonal elements (SDIRK) and explicit first stage (ESDIRK). In each of these classes, we present new A-stable schemes of order six (the highest order of previously known A-stable DIRK-type schemes) up to order eight. For each order, we include one scheme that is only A-stable as well as schemes that are L-stable, stiffly accurate, and/or have stage order two. The latter types require more stages, but give better convergence rates for differential-algebraic equations (DAEs), and those which have stage order two give better accuracy for moderately stiff problems. The development of the eighth-order schemes requires, in addition to imposing A-stability, finding highly accurate numerical solutions for a system of 200 equations in over 100 variables, which is accomplished via a combination of global and local optimization strategies. The accuracy, stability, and adaptive stepsize control of the schemes are demonstrated on diverse problems.
