Nonlinear mixed-dimension model for embedded tubular networks with application to root water uptake

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 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.

Projection-based resolved interface mixed-dimension method for embedded tubular network systems

“…This error by far dominates the geometry-related model errors investigated in the tissue perfusion example. The result strongly suggests the need for implicit interface methods that can overcome this problem of grid resolution, for example concepts using local analytical or numerical solutions in the immediate neighborhood of the roots [17,15,16,43].…”

“…In the following, we therefore only compare the css method with the ps method. An extension of the ds method for the case of root water uptake and a comparison with the ps method is presented in [43].…”