加权多边形间断伽辽金方法用于具有三次相互作用的分层耦合反应-扩散系统
A weighted polygonal discontinuous Galerkin method for hierarchically coupled reaction-diffusion systems with cubic interactions
- Politecnico di Milano（米兰理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种对称加权内罚多面体间断伽辽金方法，用于求解具有三次非线性项的多物种反应-扩散系统，通过分层块结构和耗散对角假设实现稳定性和误差估计，并验证了其鲁棒性。
AI中文摘要：
我们开发并分析了一种对称加权内罚多面体间断伽辽金（SWIP-PolyDG）方法，用于具有高达三次阶非线性反应项的多物种反应-扩散系统。当高阶相互作用被纳入种群模型时，此类项自然出现，允许多个物种之间的非加性效应影响局部生长和转化机制。所提出的框架适用于一般多边形网格上的非均匀且可能各向异性的扩散张量。为了控制非线性耦合，我们考虑反应算子中的分层块结构，其中每个种群块仅依赖于其自身变量以及与前序块相关的变量。此外，每个块内的三次自相互作用被假定具有耗散对角结构。在这些假设下，并沿种群块的层次递归进行，我们推导了二维空间中半离散格式在$L^2(\Omega)$和dG范数下的局部时间稳定性估计。我们进一步为具有足够正则性的半离散格式解建立了组合$L^2(\Omega)$-dG能量范数下的先验误差估计。最后，数值实验证实了预测的收敛行为，并展示了该方法在非均匀扩散下的鲁棒性。
英文摘要：
We develop and analyse a symmetric weighted interior penalty polytopal discontinuous Galerkin (SWIP-PolyDG) method for multi-species reaction--diffusion systems with nonlinear reaction terms of up to cubic order. Such terms arise naturally when higher-order interactions are incorporated into population models, allowing non-additive effects among multiple species to influence local growth and conversion mechanisms. The proposed framework accommodates heterogeneous and possibly anisotropic diffusion tensors on general polygonal meshes. To control the nonlinear coupling, we consider a hierarchical block structure in the reaction operator, whereby each population block depends only on its own variables and on those associated with preceding blocks. In addition, the cubic self-interactions within each block are assumed to have a dissipative diagonal structure. Under these hypotheses and proceeding recursively over the hierarchy of population blocks, we derive local-in-time stability estimates in two spatial dimensions for the semi-discrete formulation in both the $L^2(Ω)$-and dG-norms. We further establish an a priori error estimate in a combined $L^2(Ω)$-dG energy norm for sufficiently regular solutions for the semi-discrete formulation. Finally, numerical experiments confirm the predicted convergence behaviour and illustrate the robustness of the method under heterogeneous diffusion.
