For the NIM and FIM methods, we have coded up the numerical solutions from Celia et al. (1990). This provides a typical NIM method (first-order backward Euler implicit solution, Eq. 4 in Celia et al., 1990) and two alternative FIMs: Picard iteration that solves the ψ form of RE and their improved modified Picard iteration method (MPM) solution that solves the mixed form of RE. Fully reproducible details of these models were provided by Celia et al. (1990) and so will not be repeated here. These models were implemented in Python, and the code is provided at Ireson (2022, https://github.com/amireson/openRE, last access: 24 January 2023). The NIM model was coded up in 67 lines of code (note that blank lines and comment lines are not counted in the number of lines of code), with an additional 52 lines of code to configure the problem (define the grid, hydraulic properties, etc.). The FIM Picard iteration method was coded up in 79 lines of code and required the same 52 lines of code to configure the problem. The FIM modified Picard iteration was implemented using the Numba just-in-time compiler (Sect. A5) and was coded up in 90 lines of code, with an additional 59 lines of code to configure the problem. All solutions make use of the Thomas algorithm to solve the tridiagonal linear system arising from the implicit method.

One major benefit of using a standard ODE solver to integrate RE is that the code is concise and easy to read and understand. Our objective is to write a function that will evaluate the right-hand side of Eq. (8) and feed this to the ODE solver. Within this function, the problem can be treated as instantaneous in time so that we only need to consider the spatial differences in our finite-difference solution scheme. We will use a cell-centered grid (Bear and Cheng, 2010, p. 533) in space; i.e., state variables are stored at nodes located at the center of grid cells, while fluxes are defined at the cell boundaries, as shown schematically in Fig. 1 (note that this is equivalent to what is called a “staggered grid” in fluid dynamics). For simplicity here, we consider a uniform grid (constant Δz), but it is straightforward to adapt these solutions to non-uniform grids. We introduce two spatial indices (Fig. 1): i represents the nodal values, and j represents the cell boundaries, both of which have an initial value of zero (because Python uses zero-based indexing). Hence, j=i+1/2. We will start with the ψ form of RE, for which the governing equation is given in the form 9∂ψ∂ti=-1C(ψi)∂q∂zi and 10∂q∂zi=qi+12-qi-12Δz=qj+1-qjΔz.

Here, the fluxes are given by 11qi+12=qj+1=-K(ψi+1)+K(ψi)2×ψi+1-ψiΔz-1qi-12=qj=-K(ψi)+K(ψi-1)2×ψi-ψi-1Δz-1.

In Eq. (11), we are using the arithmetic mean of K at the nodal points to estimate K at the cell boundaries, but other options are possible (see, e.g., Bear and Cheng, 2010, p. 535). It is possible to combine Eqs. (9)–(11), but keeping them separate keeps the code modular and simple.

There are three commonly used boundary conditions for this problem, namely i.

a specified flux, qT (LT-1) (type-II boundary) is often used at the upper boundary to represent infiltration, where12qj=0=qT;

Box 1 provides Python-based pseudo-code that implements this solution (Eqs. 9–14) in a function for dψ/dt with a type-II boundary at the upper boundary and a free-drainage boundary at the lower boundary and contains just seven lines of code. This function can be called by the ODE solver.

When RE is solved over some time interval, t=t0→tM (where tM-t0 typically corresponds to multiple years in practical application) for a soil profile 0≤z≤L, the cumulative inflow minus outflow should equal the change in storage in the profile over the same interval; i.e., ϵB (mm), the bias error, defined in Eq. (15), should be zero. 15ϵB=∫t=t0tM(q(t,0)-q(t,zN))dt-∫z=0L(θ(tM,z)-θ(t0,z))dz

The bias error can be treated as a mass balance performance metric for the model, but this metric may underestimate the true errors in the water balance that occur within the time period simulated, which may cancel out over the entire run. The metric used by Celia et al. (1990), Rathfelder and Abriola (1994), and Tocci et al. (1997) has the same problem. A more rigorous mass balance performance metric is the root mean squared error of the daily (or some other time increment) cumulative net flux minus the change in storage, ϵR (mm), where 16ϵR=∑j=1M∫t=tjtj+1(q(t,0)-q(t,zN))dt-∫z=0L(θ(tj+1,z)-θ(tj,z))dz2/M, where j is an index in time, and M is the number of time steps considered. Reporting both metrics is informative – a high ϵR with a low ϵB indicates that daily errors are occurring but canceling one another out; a high ϵB with a low ϵR indicates that small daily errors are systematic in one direction and hence accumulate to give a high bias.

The fluxes in the mass balance calculation depend on the boundary conditions. For type-I (specified ψ) and free-drainage-type boundary conditions, the boundary flux depends on the simulated ψ value at the node closest to the boundary. Over a time increment tj to tj+1, ψ values will change continuously and hence so will the boundary flux. Due to the non-linearity of the K(ψ) relationship, the flux will change in a non-linear manner. The cumulative flux over the time increment, Qj→j+1 (mm), is given by 17Qj→j+1=∫t=tjtj+1q(t)dt.

Qj→j+1 is estimated from discrete values of q (which in turn are approximated from discrete values of ψ, using either a forward difference approximation, where Qj→j+1≈qjΔt; a backward difference approximation (as in Celia et al., 1990), where Qj→j+1≈qj+1Δt; or a central difference approximation (as in trapezoidal integration), where Qj→j+1≈(qj+qj+1)Δt/2. These discrete approximations for Qj→j+1 can be poor if the time step is large – or, more precisely, if the changes in q over a time step are large and non-linear.

This leads to an important limitation with Celia's mixed form solution to RE (and other equivalent iterative solutions). This solution has excellent mass balance closure, but because it uses a fixed-step iterative solution procedure with a backward difference approximation for Qj→j+1, the simulated boundary fluxes can be shown to be sensitive to the time step (see Sect. 3.2). Hence, even though for larger Δt the water balance is still perfectly closed, the actual terms within the water balance have changed, so there is less inflow and less change in storage.

ATS solvers can provide a practical solution to solving RE with good mass balance performance and boundary fluxes that do not depend on the user-specified time step. The basic idea behind ATS solvers is that when there are large changes in ψ in the model, small steps can be taken to capture the shape of C(ψ) and Qj→j+1 and minimize the integration errors over a time step. When the changes in ψ are small, larger steps can be taken to maximize efficiency. We will refer to these adaptive time steps as calculation steps. The user specifies the time steps at which they wish the results to be saved, which we will refer to as the reporting time steps. In typical practical applications, the reporting step would be the resolution of the driving data (e.g., hourly or daily). The solver may take many calculation steps of varying lengths between the reporting steps, saving the outcomes internally each time that the error tolerance is satisfied and a successful step is taken.

To accurately calculate the boundary fluxes, it is necessary to use the ψ information from the calculation time steps because ψ may have evolved non-linearly over the reporting time step. However, this is not trivial. Two possible ways to do this (which are equivalent to one another) are to (i) enable dense output from the ODE solver (if this feature is supported by the ODE solver) or (ii) force the ODE solver to make the reporting steps equal to the calculation steps. However, both of these approaches are more computationally demanding in terms of memory and runtime – a significant disadvantage. We propose here a third method for calculating the boundary fluxes that can be used with any ODE solver. At an instance in time, the cumulative boundary flux, Q, is related to the instantaneous flux, q, by 18dQdt=q.

We can therefore use the ODE solver to integrate this expression and solve for Q. To do this, we add to the system of ODEs defined by Eq. (8) two new ODE expressions that represent the cumulative boundary fluxes. The dependent variable vector that is sent to the ODE solver now is F, defined 19F=QT:t0→tψ0ψ1⋮ψN-2ψN-1QB:t0→t, where QT:t0→t and QB:t0→t are the cumulative boundary fluxes at the current time, t, since the start of the simulation, t0. The ODE solver will integrate the equation 20dFdt=f(F,t,z) to solve for F. The function that is called by the ODE solver will evaluate the vector 21dFdt=dQT:t0→tjdtdψ0dtdψ1dt⋮dψN-2dtdψN-1dtdQB:t0→tjdt=qT-1C(ψ0)∂q∂z0-1C(ψ1)∂q∂z1⋮-1C(ψN-2)∂q∂zN-2-1C(ψN-1)∂q∂zN-1qB.

We note that each term in Eq. (21) is expressed at a single instant in time, t, and subscripts 0,1,…,N-2,N-1 refer to the indices of the finite-difference discretization points, and we use zero-based indexing to be consistent with the Python language and Fig. 1. After solving for F, the first and last rows of F correspond to the cumulative boundary fluxes, which at time t are QT:t0→t and QB:t0→t. The cumulative fluxes over each time step are obtained from 22QT:tj→tj+1=QT:t0→tj+1-QT:t0→tjQB:tj→tj+1=QB:t0→tj+1-QB:t0→tj.

Solving systems of ODEs in this way is straightforward – requiring the user to pack and unpack the dependent variable vector. Python-based pseudo-code showing how this can be implemented is given in Box 2. Note that here we multiply the fluxes by 1000 within the solver (equivalent to converting the fluxes from md-1 to mmd-1), so that the magnitude of the fluxes is comparable with the magnitude of the changes in ψ (typically in m), which can improve the water balance estimate. Any arbitrary scaling factor can be applied here, as long as the output fluxes are re-scaled to the correct units. We shall refer to this method for calculating the boundary fluxes as the solver flux output method (SFOM).

Here, we provide details of two techniques that can be used to improve the computational runtime of the model. These methods have no impact on the accuracy of the solution, so they are optional, but combined they can result in better than a factor of 10 reduction in the runtime at the cost of only a few additional lines of code. In Appendix A, we investigate the impact of a range of different possible model decisions/assumptions on the accuracy, mass balance, efficiency, and simplicity of the model.

Hydrus 1D (Šimůnek et al., 2005, 2016) is a widely used one-dimensional RE solver. The calculations within Hydrus are undertaken using openly available FORTRAN source code, and the software runs through a (closed-source) graphical user interface on Microsoft Windows. The FORTRAN code can be compiled using gfortran on the macOS operating system and run from the command line, which we did here, so that the runtime comparisons with our model are fair. Within Hydrus, the user interface provides somewhat limited control over the error tolerances. We were unable to modify any settings to improve the water balance, and so we present here model runs that use the default iteration criteria (maximum number of iterations = 100; water content tolerance = 0.001; pressure head tolerance = 10 mm; lower/upper optimal iteration range = 0.7/1.3; lower/upper time step multiplication factor = 1.3/0.7; lower/upper limit of tension interval = 10-6/103 cm).

We configured Hydrus 1D, our ATS solution, and our implementation of Celia's MPM solution for a simple numerical experiment, where we simulate the infiltration of a 10-year time series of daily precipitation into a 1.5 m deep soil column, with a free-drainage lower-boundary condition. The minimum, mean, and maximum annual precipitation was 265, 484, and 680 mmyr-1, and the maximum daily precipitation was 55 mmd-1. We used silt loam GE 3 soil hydraulic properties from van Genuchten (1980), where θr = 0.131, θs = 0.396, α = 0.423 m-1, Ks = 0.0496 md-1, and n=2.06. We set SS = 10-6 m-1 and used a uniform ψ initial condition of -3.59 m. The results of the simulations with each model are given in Table 5, showing the runtime and water balance performance; Fig. 6, showing the detailed water balance performance; and Fig. 7, showing the simulated storage and drainage.

In our ATS solution, we can trade off between water balance error and the runtime by modifying the rtol argument for the ODE solver. We found that the default rtol of 10-6 had the fastest runtime, but the water balance performance, whilst good enough for all practical purposes, was the worst overall (Table 5, Fig. 6). Therefore, we reduced rtol to 10-7, which improved the water balance performance but increased the runtime. Even though the water balance errors reported here are all very small, it is still interesting to look closely at how these compare for the different models, as shown in Fig. 6. The first thing to note is that the Celia MPM solution has water balance errors of essentially zero, which we expect, because this solution enforces water balance closure. Celia's solution did have the longest runtime – approximately 40 % slower than the other solutions. In the ATS solutions, on a daily basis, the water balance errors are much smaller than in Hydrus. However, in Hydrus, the water balance errors appear random, with a mean of zero, and hence when looking at the cumulative errors in Hydrus, there is no systematic accumulation in the errors. In the ATS solutions, in the lower-right panel of Fig. 6, we can see that the water balance errors are strongly correlated to the infiltration flux at the upper boundary – larger fluxes result in larger errors. Hence, for the ATS solution with rtol = 10-6, we see that the errors accumulate, and after 4 years, the cumulative errors in the ATS solution exceed those in Hydrus. For the ATS solution with rtol = 10-7, the errors do not accumulate monotonically, and the long-term cumulative errors tend to oscillate about zero. The water balance performance of the ATS solution with rtol = 10-7 is therefore better than the performance in Hydrus (Table 5, Fig. 6), while the runtimes of these models are essentially the same (Hydrus is slightly faster, with a runtime of 2.21 vs. 2.30 s, Table 5).

Looking at the simulated storage and discharge in Fig. 7, the two ATS solutions are visually indistinguishable and are both broadly consistent with the Hydrus 1D model outputs. The Celia MPM solution has non-negligible differences with all other solutions. This is because the MPM solution applies an iterative solution procedure to solve the model at a daily time step, and the boundary fluxes are therefore subject to errors, as discussed above. The solution scheme imposes mass balance on the problem but does not track the truncation errors in the fluxes. The avoidance of this issue represents a significant advantage of adaptive time-stepping solutions.