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.
A Stiff Order Condition Theory for Runge-Kutta Methods Applied to Semilinear ODEs
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；New Jersey Institute of Technology（新泽西理工学院）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Temple University（天普大学）
Efficient high order schemes for stiff ODEs in cardiac electrophysiology
INRIA Bordeaux Sud Ouest, Université de Bordeaux（法国国家信息与自动化研究所波尔多西南分部，波尔多大学）；Institut de Mathématiques de Bordeaux, UMR CNRS 5241（波尔多数学研究所）；Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 5142, Université de Pau et des Pays de l’Adour（波尔多大学及阿杜尔地区大学数学及其应用实验室）
Adaptive Diagonally Implicit Runge-Kutta Methods Devoid of Order Reduction for Semilinear ODEs
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；New Jersey Institute of Technology（新泽西理工学院）；Temple University（天普大学）
Exponential Adams Bashforth integrators for stiff ODEs, application to cardiac electrophysiology
INRIA Bordeaux, Institut de Mathématiques de Bordeaux, UMR CNRS 5241, Université de Bordeaux（法国国家信息与自动化研究所波尔多中心，波尔多数学研究所）；Université de Pau et des Pays de l’Adour（波尔多大学）；Laboratoire de Mathématiques et de leurs Applications, UMR CNRS 5142, Université de Pau et des Pays de l’Adour（波尔多大学）
A physics-informed neural network method for the approximation of slow invariant manifolds for the general class of stiff systems of ODEs
Modelling Engineering Risk and Complexity, Scuola Superiore Meridionale（工程风险与复杂性建模，南方高级学校）；Institute of Science and Technology for Energy and Sustainable Mobility, Consiglio Nazionale delle Ricerche（能源与可持续移动科技研究所，国家研究理事会）
Proximal Implicit ODE Solvers for Accelerating Learning Neural ODEs
University of Utah（犹他大学）；University of California, Los Angeles（加州大学洛杉矶分校）；Illinois Institute of Technology（伊利诺伊理工学院）；University of California, Irvine（加州大学欧文分校）；Scientific Computing and Imaging (SCI) Institute（科学计算与成像研究所）
