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.

翻译：采用数值搜索方法设计了配备嵌入式误差估计器的高阶对角隐式Runge-Kutta (DIRK)格式，其中部分格式具有相同的对角元素(SDIRK)和显式第一级(ESDIRK)。针对每类格式，我们提出了从六阶（此前已知A-稳定DIRK型格式的最高阶数）到八阶的新A-稳定格式。对于每个阶数，我们既包含仅具有A-稳定性的格式，也包含具有L-稳定、刚性精确和/或二级阶数的格式。后几种格式需要更多级，但能对微分代数方程(DAE)提供更好的收敛速率，而具有二级阶数的格式对中等刚性问题的精度更优。开发八阶格式时，除施加A-稳定性条件外，还需对含100多个变量的200个方程组成的系统求取高精度数值解，这通过全局与局部优化策略相结合的方式实现。通过多种算例验证了这些格式的精度、稳定性和自适应步长控制能力。