[论文解读] Variational Integrators in Plasma Physics

M. Kraus|arXiv (Cornell University)|Jul 22, 2013

Numerical methods for differential equationsMathematics参考文献 6被引用 21

一句话总结

该论文通过先对拉格朗日量和变分原理进行离散化，为离子体物理模型——导心动力学、Vlasov-Poisson方程和磁流体动力学——开发了变分积分器，确保能量、动量及其他不变量在机器精度范围内精确守恒。该方法通过引入Ibragimov的积分因子，扩展至无标准拉格朗日量的系统，实现了辛的、几何结构保持的时间积分器，可防止数值耗散和非物理解的能量损失。

ABSTRACT

Variational integrators are a special kind of geometric discretisation methods applicable to any system of differential equations that obeys a Lagrangian formulation. In this thesis, variational integrators are developed for several important models of plasma physics: guiding centre dynamics (particle dynamics), the Vlasov-Poisson system (kinetic theory), and ideal magnetohydrodynamics (plasma fluid theory). Special attention is given to physical conservation laws like conservation of energy and momentum. Most systems in plasma physics do not possess a Lagrangian formulation to which the variational integrator methodology is directly applicable. Therefore the theory is extended towards nonvariational differential equations by linking it to Ibragimov's theory of integrating factors and adjoint equations. It allows us to find a Lagrangian for all ordinary and partial differential equations and systems thereof. Consequently, the applicability of variational integrators is extended to a much larger family of systems than envisaged in the original theory. This approach allows for the application of Noether's theorem to analyse the conservation properties of the system, both at the continuous and the discrete level. In numerical examples, the conservation properties of the derived schemes are analysed. In case of guiding centre dynamics, momentum in the toroidal direction of a tokamak is preserved exactly. The particle energy exhibits an error, but the absolute value of this error stays constant during the entire simulation. Therefore numerical dissipation is absent. In case of the kinetic theory, the total number of particles, total linear momentum and total energy are preserved exactly, i.e., up to machine accuracy. In case of magnetohydrodynamics, the total energy, cross helicity and the divergence of the magnetic field are preserved up to machine precision.

研究动机与目标

• 为离子体物理模型开发保持结构的数值格式，精确守恒能量、动量和磁通量等关键物理不变量。
• 通过引入Ibragimov的积分因子理论和伴随方程，将变分积分器理论扩展至不具有标准拉格朗日量的系统。
• 通过保持辛结构和动量守恒等几何结构，确保在粒子、动力学和流体离子体模型中实现长期数值稳定性和精度。
• 证明离散变分积分器可实现无条件稳定的模拟，且无任何人工耗散，即使在长时间积分下亦然。

提出的方法

• 首先对连续拉格朗日量和作用量原理进行离散化，然后通过离散变分原理推导离散运动方程，确保内在的守恒性质。
• 在连续和离散两个层次上应用诺特定理，识别能量、动量和交叉螺旋度等守恒量。
• 利用Ibragimov方法为Vlasov-Poisson和MHD等系统构造拉格朗日量，这些系统原本不自然地具有标准拉格朗日量形式。
• 对Vlasov-Poisson系统采用基于网格的离散Nambu括号形式，利用泛函导数和对称化泊松括号，以保持反对称性和守恒律。
• 在时空单元上构造离散作用量泛函 $ \mathcal{A}_d[\varphi] = \sum_{\square} \mathcal{L}_d \circ j^1\varphi(\square) $，以推导一致的时间推进格式。
• 通过使用所有偶排列和奇排列对离散Nambu括号进行对称化，以在离散层面上保持反对称性和几何一致性。

实验结果

研究问题

• RQ1 能否为缺乏标准拉格朗日量形式的离子体模型（如Vlasov-Poisson和MHD系统）构造变分积分器？
• RQ2 如何将离散变分原理应用于保持能量、动量和磁场散度等守恒律，以实现离子体模拟中的结构保持？
• RQ3 与标准时间积分器相比，变分积分器在多大程度上能防止数值耗散和虚假能量损失？
• RQ4 通过Ibragimov理论的扩展拉格朗日量框架，能否系统地应用于推导一般离子体系统的结构保持格式？

主要发现

• 在导心动力学中，角动量精确守恒，粒子能量误差在长时间模拟中保持有界，且无数值耗散。
• 在Vlasov-Poisson系统中，总粒子数、线性动量和能量精确守恒至机器精度，无任何人工损失或增益。
• 在磁流体动力学中，总能量、交叉螺旋度和磁场散度均精确守恒至机器精度，确保模拟稳定且物理解一致。
• 离散Nambu括号形式重现了Arakawa的离散化结果，表明变分方法与哈密顿方法在结构保持方面具有一致性。
• 对称化后的离散Nambu括号确保了反对称性和几何一致性，这对于在动力系统中保持底层泊松结构至关重要。