Abstract
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 analyze 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.
Highlights
Nonlinear elliptic equations arise in the description of fluid flow in porous media where permeability depends on the water pressure (e.g. Richards’ equation) or the description of heat conduction where the thermal conductivity depends
This close match between the novel distribution kernelbased method (DS) method and the PS method using a fully-resolved rootsoil interface and more than a thousand times more mesh cells is remarkable. This shows that the root water uptake problem, in particular in drier soils is clearly dominated by large and very localized gradients at the root-soil interface that are difficult to approximate for standard numerical schemes
Mixed-dimensional methods are efficient methods for solving coupled mixeddimensional PDEs arising from flow and transport processes in systems with tubular network systems embedded in a bulk domain
Summary
Nonlinear elliptic equations arise in the description of fluid flow in porous media where permeability depends on the water pressure (e.g. Richards’ equation) or the description of heat conduction where the thermal conductivity depends. We discuss a numerical scheme to solve such equations in the presence of an embedded thin tubular transport system exchanging mass or energy with the embedding bulk domain This exchange is modeled by local source terms and results in coupled systems of mixed-dimensional partial differential equations. The water distribution around a three-dimensional root network taking up water from the embedding soil can be modeled by Eq (1.1) [3, 4, 5, 1] In this case, the unknowns are hydraulic pressures in root and soil, and Db corresponds to the hydraulic conductivity.
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