标题： 完全非线性凝聚-分裂模型的数值格式，来自波动力学理论 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数值方法，用于研究完全非线性凝聚-分裂模型，其中碰撞算子的凝聚和分裂部分都是非线性的。 这些模型来源于$3-$波动力学方程，这是波湍流理论中的关键框架。 尽管波湍流理论在物理学和力学中非常重要，但针对$3-$波动力学方程的数值方案却非常少，这些方案不对解的演化施加任意的附加假设，而当前的手稿提供了一些这样的方案之一。 据我们所知，这也是第一个能够准确捕捉包含正向和反向能量级联的完全非线性凝聚-分裂模型解的长期渐近行为的数值方案。 该方案应用于一些测试问题，显示出与能量级联速率理论预测的高度一致性。 我们进一步引入了一种加权有限体积变体，以确保在不同核齐次程度下保持能量守恒。 通过理论分析建立了收敛性和一阶一致性，并通过测试案例中的实验收敛阶数进行了验证。 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.