AI 中文总结
该研究提出结合离散梯度与SAV方法的统一框架，构建适用于耗散和保守系统的保结构积分器，包含三种不同特性的积分器，可扩展至近泊松系统并保留卡西米尔不变量，通过四类数值实验验证有效性。
AI 中文摘要
我们提出一种将离散梯度（DG）方法与标量辅助变量（SAV）方法相结合的框架，用于为耗散系统和保守系统构建保结构积分器。关键观察结果是，SAV二次化将动力学提升到扩展状态空间，在该空间上修正能量具有精确的离散梯度恒等式。该观点产生了三种具有不同精度和成本特性的积分器：一阶半隐式前向欧拉格式、二阶自伴随中点格式，以及隐式度降低的二阶预测格式。该构造可扩展到近泊松系统，并在可执行离散条件下保留选定的卡西米尔不变量。数值实验涵盖艾伦-卡恩方程、大幸-川崎型非局部梯度流、双阱哈密顿振子，以及具有非线性三次卡西米尔的泊松系统。
英文摘要
We present a framework combining discrete gradient (DG) methods with the Scalar Auxiliary Variable (SAV) approach to construct structure-preserving integrators for dissipative and conservative systems. The key observation is that SAV quadratization lifts the dynamics to an extended state space on which the modified energy has an exact discrete-gradient identity. This viewpoint yields three integrators with different accuracy and cost profiles: a first-order semi-implicit Forward Euler scheme, a second-order self-adjoint Midpoint scheme, and a second-order Predictive scheme with reduced implicitness. The construction extends to almost-Poisson systems and preserves selected Casimir invariants under an enforceable discrete condition. Numerical experiments cover the Allen--Cahn equation, an Ohta--Kawasaki-type nonlocal gradient flow, a double-well Hamiltonian oscillator, and a Poisson system with a nonlinear cubic Casimir.
Comments32 pages, 7 figures
