Comparison of numerical methods for radial solute transport to simulate uptake by plant roots

The 1D radial solute transport model with non-linear inner boundary condition is widely used for simulating nutrient uptake by plant roots. When included into an architectural root model, this local model has to be solved for a high number of root segments, e. g. – segments for large root systems. Each root segment comes with its own local parameter set in heterogeneous root architectural models. Depending on the soil and solute, the effective diffusion coefficient spans over more than six orders (e. g. for N, K, and P). Thus a numerical implementation of this rhizosphere transport model is required to be fast, accurate and stable for a large parameter space. We apply 13 methods to this rhizosphere model with root hairs and compare their accuracy, computational speed, and applicability. In particular, the Crank-Nicolson method is compared to higher-order explicit adaptive methods and some stiff solvers. The Crank-Nicolson method sometimes oscillated and was up to a hundred times slower than an explicit adaptive scheme with similar accuracy. For a given spatial resolution, Crank-Nicolson had about one order lower accuracy as other tested methods. The maximum spatial time step can be estimated from root radius, solute diffusion, advection, and soil buffer power. Although Crank-Nicolson is a viable method and often used as de-facto standard method for rhizosphere models, it was not the best performer in our comparison. While the best method remains problem specific, for general use in root architectural models we recommend adaptive Runge-Kutta with cubic or quadratic upwind for advection