Rusteau-1D Help
Theory and equations

Numerical solution

Rusteau uses the numerical methods of HYDRUS-1D Šimůnek et al. (2009): Galerkin-type linear finite elements in space and finite differences in time.

Water flow

The Richards equation is discretized with linear finite elements on the nodes of the profile, with a lumped (diagonal) mass matrix, and with a fully implicit finite difference scheme in time Vogel (1987). The non-linearity is resolved by the mass-conservative modified Picard iteration of Celia et al. (1990): the time derivative is written in terms of the water content,

(θij+1,k+1 − θij) / Δt = Cij+1,k (hij+1,k+1 − hij+1,k) / Δt + (θij+1,k − θij) / Δt

(j time level, k iteration), which conserves mass much better than the head form. Each iteration solves a tridiagonal system (Thomas algorithm). The iteration ends when the changes of θ (unsaturated nodes) and h (saturated nodes) are below the tolerances (iteration criteria). The first estimate of the heads of a new time step is extrapolated from the two previous time levels.

Time step control

Three time discretizations are used: one for the numerical solution, one for the boundary conditions (the records) and one for the output (the print times). The time step of the solution adapts automatically:

  1. it always ends at the times of the boundary records and the print times;
  2. it stays between Δtmin and Δtmax;
  3. after a step that converged in at most ItMin iterations (3) it is multiplied by dMul (1.3), after a step with at least ItMax iterations (7) by dMul2 (0.7);
  4. when the iteration does not converge within MaxIt iterations, the step is repeated with Δt/3; below Δtmin the run stops.

The solute and heat transport can further limit the step (Courant number, see discretization). As in HYDRUS, the time step is handled in single precision so that the time levels agree with HYDRUS.

Interpolation tables

To save time, θ(h), K(h) and C(h) of every material are tabulated at 100 logarithmically spaced heads between hTab1 and hTabN and interpolated linearly in log |h| Vogel (1987); outside this interval the functions are evaluated directly (iteration criteria).

Solute and heat transport

The advection–dispersion and heat transport equations are solved after the water flow of each time step with Galerkin linear finite elements and a weighted time scheme (ε = 0.5: Crank–Nicolson; 1: implicit) Huyakorn and Pinder (1983). With non-linear sorption the solution is iterated. To limit oscillations at sharp fronts, upstream weighting Christie et al. (1976) or artificial dispersion can be used: the artificial dispersion is added where the product of the Peclet and Courant numbers exceeds the stability criterion Perrochet and Berod (1993)

Pe · Cr ≤ ωs = 2

Mass balance

At every print time the water and solute balance of the whole profile and of the sub-regions is computed from the nodal values and the cumulative boundary fluxes (mass balance); its error measures the accuracy of the numerical solution.

Literature cited in this topic

  • Celia, M. A., E. T. Bouloutas, and R. L. Zarba (1990). A general mass-conservative numerical solution for the unsaturated flow equation. Water Resources Research 26(7): 1483–1496. doi:10.1029/WR026i007p01483
  • Christie, I., D. F. Griffiths, A. R. Mitchell, and O. C. Zienkiewicz (1976). Finite element methods for second order differential equations with significant first derivatives. International Journal for Numerical Methods in Engineering 10(6): 1389–1396. doi:10.1002/nme.1620100617
  • Huyakorn, P. S., and G. F. Pinder (1983). Computational Methods in Subsurface Flow. Academic Press, New York, 473 pp.
  • 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
  • Vogel, T. (1987). SWMII – Numerical Model of Two-Dimensional Flow in a Variably Saturated Porous Medium. Research Report No. 87, Department of Hydraulics and Catchment Hydrology, Agricultural University, Wageningen, The Netherlands.
  • 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.