耗散哈密顿系统的四阶能量一致平均向量场离散梯度方法

中文摘要 AI 辅助

离散梯度方法为耗散哈密顿系统的能量一致时间积分提供了一种保结构途径。然而，在其传统形式下，它们通常至多达到二阶收敛。将先前为泊松系统发展的扰动原理推广到耗散情形，我们推导出四阶平均向量场离散梯度格式。一种基于同余的补全保持了耗散矩阵的正半定性，从而对所有正步长保证无条件能量耗散性。该补全概念为高阶能量一致格式提供了一条系统路径。对于纯耗散系统，推导了一种替代的指数补全方法，以及保结构的有理逼近。特别地，我们刻画了保持正半定性并可纳入能量相关分裂格式的Padé逼近。在阻尼物理摆和耗散费米-帕斯塔-乌拉姆系统上的数值实验表明，正半定补全对于避免非物理数值不稳定性和捕捉正确的定性动力学至关重要。实验进一步表明，所提出的方法在计算成本上与现有Galerkin型方法相当的情况下，实现了所需的精度和能量一致性。此外，在所考虑的精度范围内，尽管四阶格式具有更高的每步成本，但其计算效率优于传统的二阶离散梯度方法。

英文摘要

Discrete gradient methods provide a structure-preserving approach to energy-consistent time integration of dissipative Hamiltonian systems. In their conventional formulation, however, they generally attain at most second-order convergence. Extending the perturbation principle previously developed for Poisson systems to the dissipative setting, we derive fourth-order average vector field discrete gradient schemes. A congruence-based completion preserves the positive semi-definiteness of the dissipation matrix, thereby guaranteeing unconditional energy dissipativity for all positive step sizes. This completion concept provides a systematic route to higher-order energy-consistent schemes. For purely dissipative systems, an alternative exponential completion is derived, together with structure-preserving rational approximations. In particular, we characterize Padé approximants that preserve positive semi-definiteness and can be incorporated into energy-associated splitting schemes. Numerical experiments on a damped physical pendulum and a dissipative Fermi--Pasta--Ulam system demonstrate the importance of the positive semi-definite completion for avoiding nonphysical numerical instabilities and capturing the correct qualitative dynamics. They further show that the proposed methods achieve the desired accuracy and energy consistency at a computational cost competitive with existing Galerkin-type approaches. Moreover, the fourth-order schemes outperform the conventional second-order discrete gradient method in computational efficiency over the considered accuracy range, despite their higher per-step cost.

发表机构

• Trier University（特里尔大学）
• Norwegian University of Science and Technology（挪威科技大学）
• University of Wuppertal（伍珀塔尔大学）

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