We present a numerical scheme for the solution of nonlinear mixed-dimensional PDEs describing coupled processes in embedded tubular network system in exchange with a bulk domain. Such problems arise in various biological and technical applications such as in the modeling of root-water uptake, heat exchangers, or geothermal wells. The nonlinearity appears in form of solution-dependent parameters such as pressure-dependent permeability or temperature-dependent thermal conductivity. We derive and analyse a numerical scheme based on distributing the bulk-network coupling source term by a smoothing kernel with local support. By the use of local analytical solutions, interface unknowns and fluxes at the bulk-network interface can be accurately reconstructed from coarsely resolved numerical solutions in the bulk domain. Numerical examples give confidence in the robustness of the method and show the results in comparison to previously published methods. The new method outperforms these existing methods in accuracy and efficiency. In a root water uptake scenario, we accurately estimate the transpiration rate using only a few thousand 3D mesh cells and a structured cube grid whereas other state-of-the-art numerical schemes require millions of cells and local grid refinement to reach comparable accuracy.
翻译:我们提出了一个用于解决非线性混合多维PDE的数值方案,用以描述嵌入管状网络系统中以大片域交换的混合过程。这些问题出现在各种生物和技术应用中,例如根水吸收、热交换器或地热井的模型中。非线性的形式表现为依赖解决方案的参数,如依赖压力的渗透性或温度依赖热导导能。我们从一个基于在本地支持下通过平滑的内核来分配散装网络混合源术语的数字方案中得出和分析一个数字方案。通过使用本地分析解决方案,可以准确地从大片域中粗略解决的数字解决方案中重建界面的未知和通量。数字实例显示对方法的稳健性的信心,并显示与以前公布的方法相比的结果。新方法在准确和效率方面优于这些现有方法。在一种根水吸收假设中,我们精确地估计转动率,仅使用几千个3DMesh细胞和结构化的立方格,而其他状态的数值方案则需要数百万个细胞和本地网格的精确度,才能达到可比的精确度。