This article introduces a novel numerical approach for studying fully nonlinear coagulation-fragmentation models, where both the coagulation and fragmentation components of the collision operator are nonlinear. The model approximates the $3-$wave kinetic equations, a pivotal framework in wave turbulence theory governing the time evolution of wave spectra in weakly nonlinear systems. An implicit finite volume scheme (FVS) is derived to solve this equation. To the best of our knowledge, this 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 FVS 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.

翻译：本文提出了一种研究完全非线性凝聚-破碎模型的新颖数值方法，其中碰撞算子的凝聚项与破碎项均呈现非线性特征。该模型近似于$3-$波动力学方程——波湍流理论中描述弱非线性系统波谱时间演化的核心框架。我们推导了用于求解该方程的隐式有限体积格式（FVS）。据我们所知，这是首个能够精确捕捉完全非线性凝聚-破碎模型（包含正向与反向能量级串过程）解长期渐近行为的数值格式。该格式在若干测试问题中得以实施，结果显示其与能量级串速率的理论预测高度吻合。我们进一步提出了加权FVS变体，以确保在不同核函数齐次性程度下的能量守恒特性。通过理论分析建立了格式的收敛性与一阶相容性，并在测试案例中通过实验收敛阶数予以验证。