分块哈密顿系统的辛GARK方法

用于哈密顿系统的辛多速率广义加法Runge-Kutta方法

Symplectic multirate generalized additive Runge-Kutta methods for Hamiltonian systems

本研究结合辛与多速率广义加法Runge-Kutta框架，推导多尺度加法划分哈密顿系统的阶、辛性及时间可逆性条件，分析稳定性与隐式-显式积分划分，构造高阶格式并通过Fermi-Pasta-Ulam问题验证性能。

Linearly implicit GARK schemes

Systems driven by multiple physical processes are central to many areas of science and engineering. Time discretization of multiphysics systems is challenging, since different processes have different levels of stiffness and characteristic time scales. The multimethod approach discretizes each physical process with an appropriate numerical method; the methods are coupled appropriately such that the overall solution has the desired accuracy and stability properties. The authors developed the general-structure additive Runge-Kutta (GARK) framework, which constructs multimethods based on Runge-Ku…

A unified formulation of splitting-based implicit time integration schemes

Splitting-based time integration approaches such as fractional steps, alternating direction implicit, operator splitting, and locally one-dimensional methods partition the system of interest into components and solve individual components implicitly in a cost-effective way. This work proposes a unified formulation of splitting time integration schemes in the framework of general-structure additive Runge-Kutta (GARK) methods. Specifically, we develop implicit-implicit (IMIM) GARK schemes, provide the order conditions and stability analysis for this class, and explain their application to partit…

Discrete Variational Methods and Symplectic Generalized Additive Runge--Kutta Methods

We consider a Lagrangian system $L(q,\dot q) = \sum_{l=1}^{N}L^{\{l\}}(q,\dot q)$, where the $q$-variable is treated by a Generalized Additive Runge--Kutta (GARK) method. Applying the technique of discrete variations, we show how to construct symplectic schemes. Assuming the diagonal methods for the GARK given, we present some techinques for constructing the transition matrices. We address the problem of the order of the methods and discuss some semi-separable and separable problems, showing some interesting constructions of methods with non-square coefficient matrices.

Implicit multirate GARK methods

This work considers multirate generalized-structure additively partitioned Runge-Kutta (MrGARK) methods for solving stiff systems of ordinary differential equations (ODEs) with multiple time scales. These methods treat different partitions of the system with different timesteps for a more targeted and efficient solution compared to monolithic single rate approaches. With implicit methods used across all partitions, methods must find a balance between stability and the cost of solving nonlinear equations for the stages. In order to characterize this important trade-off, we explore multirate cou…

构造具有连续阶段的辛（分块）龙格-库塔方法

Construction of symplectic (partitioned) Runge-Kutta methods with continuous stage

本文研究了构造具有连续阶段的辛（分块）龙格-库塔方法，通过正交多项式展开和简化假设提出新方法，保持几何结构。