三波动力学方程的数值格式：碰撞算子的完整处理
Numerical Schemes for 3-Wave Kinetic Equations: A Complete Treatment of the Collision Operator
AI总结：
本文针对含完整复杂碰撞算子的三波动力学方程，基于新颖守恒形式构造自适应时间步进的隐式有限体积格式，实现长时间模拟，且数值解与总能量衰减的渐近估计高度一致。
摘要：
在我们此前的工作中，已成功为三波动力学方程的简化版本设计了数值格式；该简化版本仅保留碰撞算子中的简单正级联项，所设计格式尤其能够在多重爆破时间现象下捕捉方程的长时间动力学。在本系列的第二篇工作中，我们提出针对完整三波动力学方程的数值处理方法：基于该方程的一种新颖守恒形式，充分考虑完整且复杂得多的碰撞算子。随后，我们推导了一种隐式有限体积格式来求解该方程。新的离散化采用自适应时间步进方法，使模拟能够进行到极长时间。我们将计算得到的解与此前推导的、关于三波动力学方程含时解总能量衰减率的长时间渐近估计进行比较，发现二者高度吻合。
英文摘要：
In our previous work, numerical schemes for a simplified version of 3-wave kinetic equations, in which only the simple forward-cascade terms of the collision operators are kept, have been successfully designed, especially to capture the long time dynamics of the equation given the multiple blow-up time phenomenon. In this second work in the series, we propose numerical treatments for the complete 3-wave kinetic equations, in which the complete, much more complicated collision operators are fully considered based on a novel conservative form of the equation. We then derive an implicit finite volume scheme to solve the equation. The new discretization uses an adaptive time-stepping method which allows for the simulations to be carried to very long times. Our computed solutions are compared with previously derived long-time asymptotic estimates for the decay rate of total energy of time-dependent solutions of 3-wave kinetic equations and found to be in excellent agreement.
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