The solution of convection–diffusion equations for solute transport to plant roots

Abstract

To help develop models of solute uptake that are computationally efficient and suitable for up-scaling to whole root systems, we provide three alternative analytical solutions of equations for transport to absorbing roots by convection and diffusion, and we compare their performance with a numerical solution over the range of plant and soil conditions found in practice. We point out an important pitfall in commonly used methods to solve convection–diffusion equations numerically and show how it can be avoided. We identify a simple analytical solution that is valid for all realistic combinations of parameter values, and show that for most purposes it performs as well as a complete, but more complex, analytical solution allowing fully for both convection and diffusion. We conclude that this solution is suitable for use in general solute uptake models.

Appendix

Numerical solution of convection–diffusion equations

Although the diffusion part of convection–diffusion equations can be solved rapidly and efficiently using implicit finite-difference schemes, we show here that the convection part must be solved explicitly, and this sets bounds on time and space steps.

Considering the convection–diffusion equation in radial geometry (Eq. 5):

$$\frac{{\partial \,c}}{{\partial \,t}} - \frac{{Pe}}{r}\frac{{\partial \,c}}{{\partial \,r}} = \frac{1}{r}\frac{\partial }{{\partial \,r}}{\left( {r\frac{{\partial \,c}}{{\partial \,r}}} \right)}$$

(14)

the most common way of discretising this equation (see Morton and Mayers 1994 for details) uses backward finite difference approximations for the time derivative:

$$\frac{{\partial c}}{{\partial t}} \sim \frac{{U^{j}_{i} - U^{{j - 1}}_{i} }}{{\Delta t}}$$

(15)

where $c{\left( {i\Delta r,j\Delta t} \right)} = U^{j}_{i}$, and volume-conserving central difference approximations for the diffusion term:

$$\frac{1}{r}\frac{{\partial \,c}}{{\partial \,r}}{\left( {r\frac{{\partial \,c}}{{\partial \,r}}} \right)} \sim \frac{2}{{\Delta r^{2} {\left( {r_{{i + 1/2}} + r_{{i - 1/2}} } \right)}}}{\left[ {r_{{i + 1/2}} U^{j}_{{i + 1}} - {\left( {r_{{i + 1/2}} + r_{{i - 1/2}} } \right)}\,U^{j}_{i} + r_{{i - 1/2}} U^{j}_{{i - 1}} } \right]}$$

(16)

Discretisation of the convection term is more difficult. The convection term is hyperbolic and because convection only occurs in one direction – unlike diffusion – the solution at the following time step depends only on the values within the domain of influence of the previous time step. Hence, for Pe > 0, the convection term must be discretised with forward difference approximations (the ‘upwind’ scheme – Morton and Mayers 1994 Chapter 4 and page 91), i.e.,

$$\frac{{Pe}}{r}\frac{{\partial \,c}}{{\partial \,r}} \sim \frac{{Pe}}{{r_{i} }}\frac{{U^{{j - 1}}_{{i + 1}} - U^{{j - 1}}_{i} }}{{\Delta r}}$$

(17)

To ensure that the points at the previous time step lie within the domain of influence of the hyperbolic term, the Courant–Friedrichs–Lewy condition must be satisfied (Morton and Mayers 1994). In our case this means that there is a restriction on the time and space discretisation that depends on the value of the Peclet number and there is a condition for stability given by

$$Pe\Delta t < \Delta r$$

(18)

Thus, the time step must be decreased as Pe increases to ensure the stability of the upwind scheme. The upwind scheme given by Eq. 17 is the simplest form and is known to have problems with stability if the solution is likely to have sharp interfaces. In this case we recommend the Lax-Wendorf, leap-frog and other more advanced schemes designed for convection terms (see Morton and Mayers (1994) for details).

In summary, the discertisation given by Eqs 14, 15, 16, and 17 results in the following tridiagonal system for U _i ^j:

$$\begin{aligned} & - a_{i} U^{j}_{{i - 1}} + b_{i} U^{j}_{i} - c_{i} U^{j}_{{i + 1}} = d_{i} \\ & a_{i} = \frac{{2r_{{i - 1/2}} \Delta t}}{{{\left( {r_{{i + 1/2}} + r_{{i - 1/2}} } \right)}\Delta r^{2} }} \\ & c_{i} = \frac{{2r_{{i + 1/2}} \Delta t}}{{{\left( {r_{{i + 1/2}} + r_{{i - 1/2}} } \right)}\Delta r^{2} }} \\ & b_{i} = 1 + a_{i} + c_{i} \\ & d_{i} = U^{{j - 1}}_{i} + \frac{{\Delta t}}{{\Delta r}}\frac{{Pe}}{{r_{i} }}{\left( {U^{{j - 1}}_{{i + 1}} - U^{{j - 1}}_{i} } \right)} \\ \end{aligned}$$

(19)

The boundary conditions at the root surface and far away from the root surface are incorporated in this scheme in the usual way. Thus, at i = 0 we have

$$\frac{{U^{j}_{1} - U^{j}_{{ - 1}} }}{{2\Delta r}} + PeU^{{j - 1}}_{0} = \frac{\lambda }{{1 + U^{{j - 1}}_{0} }}U^{j}_{0} - \varepsilon $$

(20)

and solving this together with Eq. 18 evaluated at i = 0 gives

$${\left( {a_{0} \frac{{2\Delta x\lambda }}{{1 + U^{{j - 1}}_{0} }} + b_{0} } \right)}\,U^{j}_{0} - {\left( {a_{0} + c_{0} } \right)}\,U^{j}_{0} = d_{0} + 2\Delta ra_{0} {\left( {\varepsilon + PeU^{{j - 1}}_{0} } \right)}$$

(21)

At i = i _max we know the value of U and thus, Eqs 19 and 21 provide i _max − 1 simultaneous equations for i _max − 1 unknowns. One of the fastest ways of solving such tridiagonal matrices is to use the Thomas algorithm (see Morton and Mayers 1994, p. 24) or other linear equation solvers.