An improved iteration method for the numerical solution of groundwater flow in unsaturated soils

An improved Jacobi iterative method with two-side equilibration (TS-IJIM) is developed to model the process of unsaturated groundwater flow. The exponential model was adopted to linearize the nonlinear Richards equation (RE). Numerical discretization employing the finite difference method (FDM) produces a highly ill-conditioned system of linear equations. The TS-IJIM is proposed to reduce the matrix conditional number. Compared with two-side equilibration alone and the improved Jacobi iterative method alone, the proposed method effectively reduces the matrix conditional number, which is smallest when the resultant matrix has the same norm for each row. The TS-IJIM method is insensitive to the relaxation factor. Compared with the conjugate gradient method with a two-side equilibration algorithm, the TS-IJIM has better convergence in solving the highly ill-conditioned system. The conditional number of the ill-conditioned matrix can be reduced to 1.0 using the proposed TS-IJIM. Numerical results show that the proposed method can reduce the conditional number and improve convergence for solving unsaturated groundwater flow. Results also indicate that the proposed model is highly accurate and efficient in computing steady-state and transient 1D and 2D groundwater flow in unsaturated soils.