  Modeling unsaturated flow through soils with water uptake by plan root has
many applications in agriculture and water resources management. In this study,
our aim is to develop efficient numerical techniques for solving the Richards
equation with a sink term due to plant root water uptake. The Feddes model is
used for water absorption by plant roots, and the van-Genuchten model is
employed for capillary pressure. We introduce a numerical approach that
combines the localized exponential radial basis function (EXP-RBF) method for
space and the second-order backward differentiation formula (BDF2) for temporal
discretization. The localized RBF methods eliminate the need for mesh
generation and avoid ill-conditioning problems. This approach yields a sparse
matrix for the global system, optimizing memory usage and computational time.
The proposed implicit EXP-RBF techniques have advantages in terms of accuracy
and computational efficiency thanks to the use of BDF2 and the localized RBF
method. Modified Picards iteration method for the mixed form of the Richards
equation is employed to linearize the system. Various numerical experiments are
conducted to validate the proposed numerical model of infiltration with plant
root water absorption. The obtained results conclusively demonstrate the
effectiveness of the proposed numerical model in accurately predicting soil
moisture dynamics under water uptake by plant roots. The proposed numerical
techniques can be incorporated in the numerical models where unsaturated flows
and water uptake by plant roots are involved such as in hydrology, agriculture,
and water management.
