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.

翻译：在对某些严重问题适用时,龙格-库塔(RK)方法可能显示减少订单;虽然完全隐含的RK计划存在,避免通过高等级命令减少订单,但DIRK(间接隐含龙格-库塔)计划由于其结构简单而实际上很重要;然而,这些方法不能具有高等级秩序;低级秩序的概念也可以克服减少订单的问题,也符合DIRK结构。WSO高达3的DIRK计划过去曾根据一个简化框架提出过,但不能扩大到WSO 3。 在这一工作中,WSO的一般理论被用来克服以前的WSO屏障,并与WSO 4及以上建立实用的高等级的DIRK计划。由此产生的DIRK计划非常准确、可比较、最优化的错误系数,并证明在相关的ODE和PDE测试问题组合上表现良好。