Notes on spatial and temporal discretization

The accuracy of the numerical solution depends on the size of the finite elements and the time steps. Following the HYDRUS-1D help Šimůnek et al. (2009):

Water flow

• Use small elements where large hydraulic gradients occur: near the soil surface (infiltration, evaporation), at the interfaces of materials with very different properties, and around the water table. Elements of 0.1–1 cm near the surface of a soil with infiltration or evaporation are typical van Dam and Feddes (2000).
• Elements may grow with depth where the gradients are small; neighbouring elements should not differ in size by more than a factor of about 1.5. Use fixed points with relative element sizes on the page Soil profile.
• Infiltration into very dry, fine-textured soil forms a sharp wetting front that needs fine elements and small time steps. Soils with n close to 1 benefit from the air-entry value of −2 cm.
• Check the mass balance: relative errors above 1 % call for a finer mesh or smaller time steps.

Solute and heat transport

Two dimensionless numbers characterize the discretization of the advection–dispersion equation:

with the flux q, the element size Δx, the dispersion coefficient D, the retardation factor R and the time step Δt. Numerical oscillations are small when Pe ≤ 5 and Cr ≤ 1; oscillations can be virtually removed with Pe·Cr ≤ 2 Perrochet and Berod (1993). Rusteau, as HYDRUS, limits the time step so that the criterion Pe·Cr ≤ Pe_crit is met, and adds artificial dispersion or uses upstream weighting where it is not (solute transport: general). Both reduce oscillations at the cost of numerical dispersion: the front is smeared more than with the real dispersivity.

Practical rules:

• choose Δx at most about twice the dispersivity (Pe ≤ 2);
• the maximum Peclet and Courant numbers of a run are shown on the page Run-time information;
• with very small dispersivities (sharp fronts) refine the mesh rather than relying on artificial dispersion.

Moving fronts

Wetting fronts and solute fronts move through the profile; a mesh that is fine only near the surface may be too coarse once the front reaches deeper layers. Uniform fine meshes are safe for short profiles; for deep profiles refine where the front is expected during the period of interest.

Literature cited in this topic

• Perrochet, P., and D. Berod (1993). Stability of the standard Crank-Nicolson-Galerkin scheme applied to the diffusion-convection equation: some new insights. Water Resources Research 29(9): 3291–3297. doi:10.1029/93WR00927
• van Dam, J. C., and R. A. Feddes (2000). Numerical simulation of infiltration, evaporation and shallow groundwater levels with the Richards equation. Journal of Hydrology 233(1–4): 72–85. doi:10.1016/S0022-1694(00)00227-4
• Šimůnek, J., M. Šejna, H. Saito, M. Sakai, and M. Th. van Genuchten (2009). The HYDRUS-1D Software Package for Simulating the One-Dimensional Movement of Water, Heat, and Multiple Solutes in Variably-Saturated Media, Version 4.08. HYDRUS Software Series 3, Department of Environmental Sciences, University of California Riverside, Riverside, CA, USA, 330 pp. https://www.pc-progress.com/en/Default.aspx?h1d-description

All sources: References.