标题： 保持能量的高斯插值积分器迭代格式 显示英文标题

标题： Energy-preserving iteration schemes for Gauss collocation integrators

摘要： 在本工作中，我们为在二次哈密顿量的泊松系统高斯积分中出现的（非）线性系统开发了保持能量的迭代格式。利用高斯配点积分器与对角帕德近似之间的关系，我们建立了一个基于$Q$-Arnoldi过程的Krylov子空间迭代方案，用于线性系统，该方案不仅在收敛时保持能量——如标准迭代方案那样——而且在每个迭代步骤中也保持能量。在单次迭代步骤的精度和成本方面，它与GMRES相当，因此在给定误差容限下对高维泊松系统的时域积分具有显著的效率优势。在上述线性结果的基础上，我们考虑非线性泊松系统，并设计了用于隐式中点法则（二阶高斯积分器）的非线性求解器，利用了相关的帕德近似是一个Cayley变换这一事实。对于每个时间步长产生的非线性系统，我们提出了具有固定点和牛顿型迭代方案的算法，这些算法从它们的经典版本中继承了收敛阶数，并具有相当的成本，但具有保持能量的迭代步骤。 显示英文摘要

摘要： In this work, we develop energy-preserving iterative schemes for the (non-)linear systems arising in the Gauss integration of Poisson systems with quadratic Hamiltonian. Exploiting the relation between Gauss collocation integrators and diagonal Pad\'e approximations, we establish a Krylov-subspace iteration scheme based on a $Q$-Arnoldi process for linear systems that provides energy conservation not only at convergence --as standard iteration schemes do--, but also at the level of the individual iterates. It is competitive with GMRES in terms of accuracy and cost for a single iteration step and hence offers significant efficiency gains, when it comes to time integration of high-dimensional Poisson systems within given error tolerances. On top of the linear results, we consider non-linear Poisson systems and design non-linear solvers for the implicit midpoint rule (Gauss integrator of second order), using the fact that the associated Pad\'e approximation is a Cayley transformation. For the non-linear systems arising at each time step, we propose fixed-point and Newton-type iteration schemes that inherit the convergence order with comparable cost from their classical versions, but have energy-preserving iterates.