耗散广义流体动力学方程及其数值求解
The dissipative Generalized Hydrodynamic equations and their numerical solution
AI总结：
本文研究耗散广义流体动力学方程的数值求解，提出基于高阶时间泰勒展开的后向半拉格朗日方法及隐式/显式Runge-Kutta格式求解平流项，并比较不同方法处理扩散与碰撞源项，为含非局域非线性耦合的动力学方程提供高效高阶数值方案。
摘要：
“广义流体动力学”（GHD）代表一种描述量子物理中一维可积系统（如超冷原子或自旋链）的模型。在数学上，GHD对应于动力学类型的非线性方程，其主要未知量即统计分布函数$f(t,z,θ)$，存在于由一维位置变量$z$和一维“动力学”变量$θ$构成的相空间中，该变量实为波矢，称为“快度”。GHD方程的两个关键特征是：首先，平流项中存在非局域非线性耦合；其次，存在无穷多个守恒量，这阻止了系统热化。为突破此局限，我们考虑了耗散GHD方程，该方程通过在GHD方程右端补充非局域非线性扩散算子或玻尔兹曼型碰撞积分来获得。本文处理用于高效求解这些动力学方程的新型高阶数值方法。具体而言，我们设计了新颖的后向半拉格朗日方法求解平流部分（即Vlasov方程），该方法对平流场使用高阶时间泰勒级数展开，其各阶时间导数通过递归过程获得。平流场的这种高阶时间近似被用于设计新的隐式/显式Runge-Kutta半拉格朗日方法，并与Adams-Moulton半拉格朗日格式进行了比较。对于求解由扩散和碰撞算子构成的源项，我们使用并比较了文献中的不同数值方法。
英文摘要：
"Generalized Hydrodynamics" (GHD) stands for a model that describes one-dimensional \textit{integrable} systems in quantum physics, such as ultra-cold atoms or spin chains. Mathematically, GHD corresponds to nonlinear equations of kinetic type, where the main unknown, a statistical distribution function $f(t,z,θ)$, lives in a phase space which is constituted by a one-dimensional position variable $z$, and a one-dimensional "kinetic" variable $θ$, actually a wave-vector, called "rapidity". Two key features of GHD equations are first a non-local and nonlinear coupling in the advection term, and second an infinite set of conserved quantities, which prevent the system from thermalizing. To go beyond this, we consider the dissipative GHD equations, which are obtained by supplementing the right-hand side of the GHD equations with a non-local and nonlinear diffusion operator or a Boltzmann-type collision integral. In this paper, we deal with new high-order numerical methods to efficiently solve these kinetic equations. In particular, we devise novel backward semi-Lagrangian methods for solving the advective part (the so-called Vlasov equation) by using a high-order time-Taylor series expansion for the advection fields, whose successive time derivatives are obtained by a recursive procedure. This high-order temporal approximation of the advection fields are used to design new implicit/explicit Runge-Kutta semi-Lagrangian methods, which are compared to Adams-Moulton semi-Lagrangian schemes. For solving the source terms, constituted by the diffusion and collision operators, we use and compare different numerical methods of the literature.
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