In this work, an approximate family of implicit multiderivative Runge-Kutta (MDRK) time integrators for stiff initial value problems is presented. The approximation procedure is based on the recent Approximate Implicit Taylor method (Baeza et al. in Comput. Appl. Math. 39:304, 2020). As a Taylor method can be written in MDRK format, the novel family constitutes a multistage generalization. Two different alternatives are investigated for the computation of the higher order derivatives: either directly as part of the stage equation, or either as a separate formula for each derivative added on top of the stage equation itself. From linearizing through Newton's method, it turns out that the conditioning of the Newton matrix behaves significantly different for both cases. We show that direct computation results in a matrix with a conditioning that is highly dependent on the stiffness, increasing exponentially in the stiffness parameter with the amount of derivatives. Adding separate formulas has a more favorable behavior, the matrix conditioning being linearly dependent on the stiffness, regardless of the amount of derivatives. Despite increasing the Newton system significantly in size, through several numerical results it is demonstrated that doing so can be considerably beneficial.

翻译：本文提出了一类近似隐式多导Runge-Kutta（MDRK）时间积分器，用于求解刚性初值问题。该近似过程基于近期提出的近似隐式泰勒方法（Baeza等，Comput. Appl. Math. 39:304, 2020）。由于泰勒方法可表述为MDRK格式，本文所提出的新方法族构成了其多级推广。针对高阶导数的计算，本文研究了两种不同方案：一是直接在阶段方程中计算导数，二是在阶段方程基础上为每个导数附加独立公式。通过牛顿法线性化后发现，两种情况下牛顿矩阵的条件数存在显著差异。研究表明：直接计算方案得到的矩阵条件数对刚性高度敏感，其条件数随导数数量增加而关于刚性参数呈指数增长；而附加独立公式方案则表现出更优特性，矩阵条件数与刚性呈线性关系，且不受导数数量影响。尽管该方法会显著增大牛顿系统的规模，但多项数值实验表明，这一策略可带来显著优势。