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2.7.3.2. Numerical Solution

The equation for conservation of mass (equation 2.7.41) is integrated over each soil layer to derive a finite difference equation for the change in soil liquid water content over time (equation 2.7.66). 

The water removed by transpiration in each layer (e_i) is a function of the total transpiration (E_v^t) and the effective root fraction (r_e,i) (equations 2.7.67-2.7.70).

The soil water fluxes (q_i) are linearized using a Taylor series expansion (equations 2.7.71-2.7.72), resulting in a tridiagonal equation set (equation 2.7.73) with coefficients defined in equations 2.7.74-2.7.77.

The finite difference forms of the fluxes and partial derivatives are provided in equations 2.7.78-2.7.83, using the derivatives of soil matric potential (equations 2.7.84-2.7.86) and hydraulic conductivity (equations 2.7.87-2.7.88).

The equation sets for the top layer (i=1), intermediate layers (i=2 to N_levsoi-1), and the bottom layer (i=N_levsoi) are presented in sections 2.7.3.2.1 to 2.7.3.2.3.

An adaptive time stepping approach is used to improve the accuracy and stability of the numerical solutions, as described in section 2.7.3.2.4.