Design of DIRK Schemes with High Weak Stage Order
Design of DIRK Schemes with High Weak Stage Order
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DOI：
10.2140/camcos.2023.18.1
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发表时间：
2022-04
期刊：
影响因子：
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通讯作者：
Abhijit Biswas;D. Ketcheson;Benjamin Seibold;D. Shirokoff
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作者：
Abhijit Biswas;D. Ketcheson;Benjamin Seibold;D. Shirokoff
Runge-Kutta（RK）方法在处理某些刚性问题时可能会出现降阶现象。虽然全隐式RK格式存在，避免通过高阶阶降阶，DIRK（对角隐式龙格库塔）格式是实际上很重要的，由于其结构简单，但是，这些不能拥有高阶阶。弱阶序（WSO）的概念也可以克服降阶问题，并且与DIRK结构相容。然而，过去已经提出了WSO最多3的DIRK方案，该方案基于不能扩展到WSO 3之外的简化框架。在这项工作中，WSO的一般理论，以克服现有的WSO障碍，并构造实用的高阶DIRK格式与WSO 4及以上。所得到的DIRK计划是严格准确的，L-稳定的，具有优化的误差系数，并表现出良好的相关ODE和PDE测试问题的投资组合。
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.