Taylor series methods show a newfound promise for the solution of non-stiff ordinary differential equations (ODEs) given the rise of new compiler-enhanced techniques for calculating high order derivatives. In this paper we detail a new Julia-based implementation that has two important techniques: (1) a general purpose higher-order automatic differentiation engine for derivative evaluation with low overhead; (2) a combined symbolic-numeric approach to generate code for recursively computing the Taylor polynomial of the ODE solution. We demonstrate that the resulting software's compiler-based tooling is transparent to the user, requiring no changes from interfaces required to use standard explicit Runge-Kutta methods, while achieving better run time performance. In addition, we also developed a comprehensive adaptive time and order algorithm that uses different step size and polynomial degree across the integration period, which makes this implementation more efficient and versatile in a broad range of dynamics. We show that for codes compatible with compiler transformations, these integrators are more efficient and robust than the traditionally used explicit Runge-Kutta methods.
Fast symplectic integrator for Nesterov-type acceleration method
Center of Mathematics for Artificial Intelligence and Data Science（人工智能与数据科学数学中心）；Chubu University（中部大学）；The Institute of Statistical Mathematics（统计数学研究所）；RIKEN AIP（理化学研究所先进智能项目中心）
Personalized Algorithm Generation: A Case Study in Learning ODE Integrators
National University of Singapore（新加坡国立大学）；TU München（慕尼黑工业大学）；Johns Hopkins University（约翰斯·霍普金斯大学）；Forschungszentrum Jülich GmbH（于利希研究中心）；RWTH Aachen University（亚琛工业大学）
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）
Local Linearization-Runge Kutta Methods: a class of A-stable explicit integrators for dynamical systems
IMPA（巴西数学研究所）；Universidad de Ciencias Informáticas（古巴信息科学大学）；CIMFAV-DEUV, Facultad de Ciencias, Universidad de Valparaiso（瓦尔帕莱索大学理学院）；Instituto de Cibernética, Matemática y Física（古巴计算机、数学与物理研究所）；Montreal Neurological Institute, McGill University（麦吉尔大学蒙特利尔神经学研究所）
Numerical integrators for continuous disordered nonlinear Schrödinger equation
本文研究了含有空间随机势的连续无序非线性薛定谔方程的数值解。我们解决有限时间精度降阶问题的通常的数值积分在这个问题上，这是由于随机/粗糙的潜力的存在。通过使用最近提出的来自（33，SIAM J. Numer.肛门，2019），我们展示了如何通过损失两个空间导数来积分势项。收敛性分析表明，LRI对$H^2$中的位势具有$L^2$ -范数的二阶精度.数值实验验证了这一理论结果。更多的数值结果，以调查LRI的精度与经典方法相比，在粗糙的随机势的应用。
Improved Method for Dealing with Discontinuities in Power System Transient Simulation Based on Frequency Response Optimized Integrators Considering Second Order Derivative
Hopf algebra techniques to handle dynamical systems and numerical integrators
Konputazio Zientziak eta A. A. Saila, Informatika Fakultatea, UPV/EHU（巴斯克地区大学计算机学院）；Departamento de Matemáticas, Universidad Carlos III de Madrid（马德里卡洛斯三世大学数学系）
