基于流平均的刚性ODE、SDE和具有隐藏慢动力学的哈密顿系统的非侵入式保结构多尺度积分
Non-intrusive and structure preserving multiscale integration of stiff ODEs, SDEs and Hamiltonian systems with hidden slow dynamics via flow averaging
AI总结：
提出一类基于流平均的多尺度积分器，无需显式分离快慢变量，具有非侵入性、双尺度收敛性和保结构特性，适用于刚性ODE、SDE及哈密顿系统。
摘要：
我们引入了一类新的刚性常微分方程和随机微分方程积分器。这些积分器具有以下特点：(i) 多尺度：基于流平均，不完全解析快变量，计算代价由慢变量决定；(ii) 多功能：方法基于给定动力系统（可能包含隐藏的快慢过程）的流平均，而非假设分离的快慢过程的瞬时漂移平均，从而避免显式（或数值）识别快慢变量；(iii) 非侵入性：已有的微观时间尺度数值方案可作为黑箱使用，通过在微观时间尺度上开启大系数并在介观时间尺度上关闭，轻松转化为本文的积分器之一；(iv) 双尺度收敛：在慢过程上强收敛，在快过程上测度意义下收敛。我们引入了相关的双尺度流收敛概念，并分析了这些积分器在诱导拓扑下的收敛性；(v) 保结构：对于刚性哈密顿系统（可能在流形上），可使其在所有变量上辛、时间可逆且保持对称性（对称性是使系统不变的群作用）。它们是显式的，适用于任意刚性势（不必是二次型）。在Fermi-Pasta-Ulam问题上的应用显示了在四个数量级时间尺度上的精度和稳定性。对于刚性Langevin方程，它们是保对称、时间可逆且Boltzmann-Gibbs可逆的，在所有变量上拟辛，并在各向同性摩擦下共形辛。
英文摘要：
We introduce a new class of integrators for stiff ODEs as well as SDEs. These integrators are (i) {\it Multiscale}: they are based on flow averaging and so do not fully resolve the fast variables and have a computational cost determined by slow variables (ii) {\it Versatile}: the method is based on averaging the flows of the given dynamical system (which may have hidden slow and fast processes) instead of averaging the instantaneous drift of assumed separated slow and fast processes. This bypasses the need for identifying explicitly (or numerically) the slow or fast variables (iii) {\it Nonintrusive}: A pre-existing numerical scheme resolving the microscopic time scale can be used as a black box and easily turned into one of the integrators in this paper by turning the large coefficients on over a microscopic timescale and off during a mesoscopic timescale (iv) {\it Convergent over two scales}: strongly over slow processes and in the sense of measures over fast ones. We introduce the related notion of two-scale flow convergence and analyze the convergence of these integrators under the induced topology (v) {\it Structure preserving}: for stiff Hamiltonian systems (possibly on manifolds), they can be made to be symplectic, time-reversible, and symmetry preserving (symmetries are group actions that leave the system invariant) in all variables. They are explicit and applicable to arbitrary stiff potentials (that need not be quadratic). Their application to the Fermi-Pasta-Ulam problems shows accuracy and stability over four orders of magnitude of time scales. For stiff Langevin equations, they are symmetry preserving, time-reversible and Boltzmann-Gibbs reversible, quasi-symplectic on all variables and conformally symplectic with isotropic friction.
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