标题： 非线性凝聚-碎裂模型的数值格式，该模型源于波动力学理论 显示英文标题

标题： Numerical schemes for a fully nonlinear coagulation-fragmentation model coming from wave kinetic theory

摘要： 本文介绍了一种基于有限体积技术的新型数值方法，用于研究完全非线性凝聚-碎裂模型，其中碰撞算子的凝聚和碎裂成分均为非线性。这些模型源于$3-$波动能方程，在波湍流理论中是一个关键框架。尽管波湍流理论在物理和力学中具有重要意义，但针对$3-$波动能方程的数值方案却寥寥无几，且没有对解的演化施加额外假设。本文所提供的方案之一正是其中之一。据我们所知，这也是首个能够准确捕捉包含前向和后向能级级联的完全非线性凝聚-碎裂模型解的长期渐近行为的数值方案。该方案在一些测试问题上得以实现，并显示出与能量级联率理论预测的高度一致性。此外，我们进一步引入了一种加权有限体积变体，以确保不同核齐次度下能量守恒。收敛性和一阶相容性通过理论分析得到证明，并通过实验收敛阶在测试案例中验证。 显示英文摘要

摘要： This article introduces a novel numerical approach, based on Finite Volume Techniques, for studying fully nonlinear coagulation-fragmentation models, where both the coagulation and fragmentation components of the collision operator are nonlinear. The models come from $3-$wave kinetic equations, a pivotal framework in wave turbulence theory. Despite the importance of wave turbulence theory in physics and mechanics, there have been very few numerical schemes for $3-$wave kinetic equations, in which no ad-hoc additional assumptions are imposed on the evolution of the solutions, and the current manuscript provides one of the first of such schemes. To the best of our knowledge, this also is the first numerical scheme capable of accurately capturing the long-term asymptotic behavior of solutions to a fully nonlinear coagulation-fragmentation model that includes both forward and backward energy cascades. The scheme is implemented on some test problems, demonstrating strong alignment with theoretical predictions of energy cascade rates. We further introduce a weighted Finite Volume variant to ensure energy conservation across varying degrees of kernel homogeneity. Convergence and first-order consistency are established through theoretical analysis and verified by experimental convergence orders in test cases.