Pierluigi Amodio, Luigi Brugnano, Felice Iavernaro
In recent years, the class of energy-conserving methods named Hamiltonian Boundary Value Methods (HBVMs) has been devised for numerically solving Hamiltonian problems. In this short note, we study their natural formulation as continuous-stage Runge-Kutta(-Nyström) methods, which allows a deeper insight in the methods.
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Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their connections with Runge-Kutta collocation methods
研究哈密顿边值方法(HBVMs)在龙格-库塔形式下具有与高斯-勒让德方法相同谱（除零特征值外）的等谱性质，从而证明其完美A稳定性，并阐明与经典龙格-库塔配置方法的联系。
Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving Runge-Kutta methods for the numerical solution of polynomial Hamiltonian systems
本文引入并分析了一类名为哈密顿边值方法（HBVMs）的数值方法，能够精确保持任意高次多项式哈密顿函数，具有对称性、A-稳定性和任意高阶精度。
The parametric instability arising when ordinary differential equations (ODEs) are numerically integrated with Runge-Kutta-Nyström (RKN) methods with varying step sizes is investigated. It is shown that when linear constant coefficient ODEs are integrated with RKN methods that are based on A-stable Runge-Kutta methods, the solution is nonincreasing in some norm for all positive step sizes, constant or varying. Perturbation methods are used to quantify the critical step sizes associated with parametric instability.
A Phase-Fitted Runge-Kutta-Nyström method for the Numerical Solution of Initial Value Problems with Oscillating Solutions
提出一种基于Dormand和Prince四阶代数精度Runge-Kutta-Nyström方法的无限阶相位滞后新方法，用于求解二阶周期初值问题，数值实验表明新方法比经典方法更高效。
Enhanced HBVMs for the numerical solution of Hamiltonian problems with multiple invariants
本文提出增强型HBVMs，用于求解具有多重不变量的哈密顿问题，通过理论分析和数值测试验证其有效性。
Different families of Runge-Kutta-Nyström (RKN) symplectic splitting methods of order 8 are presented for second-order systems of ordinary differential equations and are tested on numerical examples. They show a better efficiency than state-of-the-art symmetric compositions of 2nd-order symmetric schemes and RKN splitting methods of orders 4 and 6 for medium to high accuracy. For some particular examples, they are even more efficient than extrapolation methods for high accuracies and integrations over relatively short time intervals.
In recent years, the efficient numerical solution of Hamiltonian problems has led to the definition of a class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Such methods admit an interesting interpretation in terms of continuous-stage Runge-Kutta methods, which is here recalled and revisited for general differential problems.
Functionally fitted Runge-Kutta-Nyström methods
本文提出功能适配的Runge-Kutta-Nyström方法，通过格点框架分析其精确积分能力和超收敛特性，探讨了稳定性分析。
Symmetric integrators based on continuous-stage Runge-Kutta-Nystrom methods for reversible systems
本文基于连续级 Runge-Kutta-Nystrom 方法和 Legendre 展开技术，构造了求解二阶常微分方程的对称积分器，并通过数值实验验证了其有效性。
Symplecticity-preserving continuous-stage Runge-Kutta-Nyström methods
本文提出保辛的连续阶段Runge-Kutta-Nyström方法，用于求解二阶常微分方程。通过Legendre多项式展开，给出构造保辛RKN方法的简单方法。
Automated Runge-Kutta-Nyström time stepping for finite element methods in Irksome
本文介绍 Irksome 库新增的 Runge-Kutta-Nyström 自动化时间步进功能，可直接离散 PDE 的二阶时间导数，避免引入辅助变量，从而减小代数系统规模并提升效率，数值实验表明其在波动方程和孔隙弹性问题中具有竞争力。
Functionally-fitted explicit pseudo two-step Runge-Kutta-Nyström methods
本文研究了用于求解二阶初始值问题的功能适配显式伪两步Runge-Kutta-Nyström方法，证明了其阶数和阶段阶数，并展示了其在特定问题上的优越性。
Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their blended implementation
本文研究Hamiltonian边界值方法(HBVMs)在作为Runge-Kutta方法时，其Butcher表矩阵与对应Gauss-Legendre方法具有相同非零谱的性质，从而保证方法始终完美A稳定，并支持高效的混合实现。
Continuous-stage Runge-Kutta-NystrÖm methods
本文提出连续阶段Runge-Kutta-Nyström方法，通过引入加权函数并修改连续阶段方法的原始模式，建立了一个新的更大框架，从而推导出更有效的RKN型方法，并应用经典加权正交多项式构造对称和辛积分器。
A class of explicit pseudo two-step Runge-Kutta-Nyström (GEPTRKN) methods for solving second-order initial value problems $y'' = f(t,y,y')$, $y(t_0) = y_0$, $y'(t_0)=y'_0$ has been studied. This new class of methods can be considered a generalized version of the class of classical explicit pseudo two-step Runge-Kutta-Nyström methods. %The new methods will be denoted by GEPTRKN methods. We proved that an $s$-stage GEPTRKN method has step order of accuracy $p=s$ and stage order of accuracy $r=s$ for any set of distinct collocation parameters $(c_i)_{i=1}^s$. Super-convergence for order of accura…
Energy-preserving continuous-stage Runge-Kutta-Nyström methods
针对二阶动力学系统传统积分器不能保持能量等不变量的问题，本文构造保能量 csRKN 方法，给出充分条件并证明其可由连续级分块 Runge-Kutta 方法导出，辅以数值验证。
We generalize previous work by Mardal, Nilssen, and Staff (2007, SIAM J. Sci. Comp. v. 29, pp. 361-375) and Rana, Howle, Long, Meek, and Milestone (2021, SIAM J. Sci. Comp. v. 43, p. 475-495) on order-optimal preconditioners for parabolic PDEs to a larger class of differential equations and methods. The problems considered are those of the forms $u_{t}=-\mathcal{K}u+g$ and $u_{tt}=-\mathcal{K}u+g$, where the operator $\mathcal{K}$ is defined by $\mathcal{K}u:=-\nabla\cdot\left(α\nabla u\right)+βu$ and the functions $α$ and $β$ are restricted so that $α>0$, and $β\ge0$. The methods considered a…
High order symplectic integrators based on continuous-stage Runge-Kutta Nystrom methods
本文提出了一种更有效的方法来构造高阶辛积分器，用于求解二阶哈密顿方程。通过利用Legendre展开来处理阶条件的简化假设，而非像之前的工作那样逐步分析阶条件，从而方便地通过截断正交级数来设计高阶辛积分器。
Multi-product splitting and Runge-Kutta-Nystrom integrators
本文通过将指数算子分裂推广为乘积和，提出多积分裂方法，并证明其与Runge-Kutta-Nyström型积分器的等价性，从而解释高阶RKN积分器仅需p-1次力评估的原因。
Functional Continuous Runge-Kutta Methods with Reuse
本文提出了一种新的连续Runge-Kutta和Runge-Kutta-Nyström方法，用于延迟泛函微分方程，通过重用步骤的最后一个阶段来提高效率，展示了三种新方法的高阶精度和更低的计算成本。
Fifth Order Runge-Kutta-Nyström Methods with Complex Coefficients
提出允许时间步长系数为复数的五阶龙格-库塔-尼斯特伦方法，重点研究具有正实部系数的方案，其误差展开主导项为纯虚数，可有效将实变量问题的阶数提高一阶。
Randomized Runge-Kutta-Nyström Methods for Unadjusted Hamiltonian and Kinetic Langevin Monte Carlo
本文提出5/2阶和7/2阶L2精确的随机化Runge-Kutta-Nyström方法，用于在未调整哈密顿和动力学Langevin Monte Carlo中近似哈密顿流，并在理论上给出精度上界，实验验证其在高维目标分布上的高效性。
Multiple invariants conserving Runge-Kutta type methods for Hamiltonian problems
本文提出增强线积分方法（ELIMs），通过扩展哈密顿BVMs和LIMs，实现哈密顿系统中多个不变量（如能量、动量等）的同时守恒，并给出数值验证。
Efficient implementation of Gauss collocation and Hamiltonian Boundary Value Methods
本文提出了一种基于Butcher矩阵分裂的高效实现策略，用于低秩保能量龙格-库塔方法（哈密顿边值方法）及其特例高斯-勒让德配置法，并验证了线性收敛性。
Recently, the efficient numerical solution of Hamiltonian problems has been tackled by defining the class of energy-conserving Runge-Kutta methods named Hamiltonian Boundary Value Methods (HBVMs). Their derivation relies on the expansion of the vector field along the Legendre orthonormal basis. Interestingly, this approach can be extended to cope with other orthonormal bases and, in particular, we here consider the case of the Chebyshev polynomial basis. The corresponding Runge-Kutta methods were previously obtained by Costabile and Napoli [33]. In this paper, the use of a different framework …
Energy conserving methods for Hamiltonian PDEs based on spectral space decomposition
本文讨论了非线性波方程数值解中的能量守恒问题，采用傅里叶展开进行空间离散后，使用HBVM类能量守恒Runge-Kutta方法求解时间离散问题。
Randomized low-rank Runge-Kutta methods
本文提出结合显式 Runge-Kutta 与随机广义 Nyström 低秩近似的新型积分器，建立误差界并实现 Rand RK4 四阶收敛，实验显示其鲁棒性和收敛速度优于现有动力学低秩方法。
High-Order Nyström/Convolution-Quadrature Solution of Time-Domain Scattering from Closed and Open Lipschitz Boundaries with Dirichlet and Neumann Boundary Conditions
本文研究了基于 Nyström 离散化的高阶卷积求积方法，用于求解二维无界区域波动方程的时域散射问题，结合线性多步法和 Runge-Kutta 方法实现了最高五阶的时间收敛精度。
Analysis and implementation of collocation methods for fractional differential equations
针对 FHBVMs 求解分数阶微分方程初值问题时缺乏线性稳定性分析的问题，本文补全该分析并改进网格选择，且已发布更新版 Matlab 代码。
High-order energy-conserving Line Integral Methods for charged particle dynamics
本文研究了用于模拟带电粒子动力学的任意高阶能量守恒方法，这些方法基于线积分方法框架推导和研究，该框架曾用于定义哈密顿边界值方法，是一种用于哈密顿问题的能量守恒龙格-库塔方法。通过一些数值测试验证了新方法的完整分析。
A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems
本文研究对称和辛扩展RKN积分器在高振荡哈密顿系统中的长期能量守恒特性，通过建立对称ERKN与三角积分器的关系及改进的调制傅里叶展开技术，证明其近似守恒性。
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