论文标题
高弱阶段秩序的DIRK计划的设计
Design of DIRK Schemes with High Weak Stage Order
论文作者
论文摘要
当应用于某些僵硬的问题时，runge-kutta（RK）方法可能会降低顺序。尽管存在完全隐式的RK方案，避免通过高阶段秩序降低秩序，但由于其结构简单性，Dirk（对角隐含的Runge-Kutta）方案实际上很重要。但是，这些不能具有较高的阶段秩序。弱阶段（WSO）的概念也可以克服降低秩序，并且与Dirk结构兼容。但是，过去提出了最多3个WSO的DIRK计划，但是，基于一个简化的框架，该框架无法扩展到WSO 3之外。在这项工作中，使用WSO的一般理论来克服先前的WSO障碍，并用WSO 4和上面构建实际上有用的高级DIRK计划。所得的DIRK方案非常准确，L稳定，具有优化的误差系数，并且被证明在相关ODE和PDE测试问题的投资组合上表现良好。
Runge-Kutta (RK) methods may exhibit order reduction when applied to certain stiff problems. While fully implicit RK schemes exist that avoid order reduction via high-stage order, DIRK (diagonally implicit Runge-Kutta) schemes are practically important due to their structural simplicity; however, these cannot possess high stage order. The concept of weak stage order (WSO) can also overcome order reduction, and it is compatible with the DIRK structure. DIRK schemes of WSO up to 3 have been proposed in the past, however, based on a simplified framework that cannot be extended beyond WSO 3. In this work a general theory of WSO is employed to overcome the prior WSO barrier and to construct practically useful high-order DIRK schemes with WSO 4 and above. The resulting DIRK schemes are stiffly accurate, L-stable, have optimized error coefficients, and are demonstrated to perform well on a portfolio of relevant ODE and PDE test problems.
