Comparison of Picard and Newton iterative schemes in the finite element modeling for one-dimensional variably saturated-unsaturated flows


  • M S Islam Department of Mathematics, Shahjalal University of Science & Technology, Sylhet-3114, Bangladesh
  • R Ahamad Department of Mathematics, Shahjalal University of Science & Technology, Sylhet-3114, Bangladesh



Richards’ equation; Numerical solution; Finite element; Picard and Newton methods; Variably saturated-unsaturated flows.


Numerical simulation of saturated-unsaturated flow in permeable media is of incredible enthusiasm for some parts of science and engineering. Demonstrating of fluid flow in variably saturated-unsaturated permeable media generally brings about systems of highly nonlinear partial differential equations, Richards’ equation, which are not reasonable diagnostically except if ridiculous and distorting presumptions are made with respect to the attributes, flow dynamics, and properties of the physical frameworks. Picard and Newton methods are the most usually practiced iterative methods for the numerical solution of the nonlinearity coupled frameworks. A finite element numerical model was developed to precisely and efficiently figure a solution of the nonlinear Richards’ equation with homogeneous and layered soil. In this work, we have considered these two regular iterative strategies which can be utilized in a solution methodology for the nonlinear Richards' equation governing flow in variably saturated-unsaturated porous media. We have evaluated the efficiency, accuracy, robustness and computational efficacy of the iterative Newton and Picard methods. The assessments depend on three distinctive test issues of one-dimensional saturated-unsaturated flow with homogeneous and heterogeneous soil properties. Besides, spatial adjustment will be founded on a fine spatial discretization and temporal adjustment will be practiced utilizing variable order, variable step size dependent on the backward Euler finite difference formula. The computed outcomes obviously demonstrated that the methodology was all the more computationally productive, yet additionally increasingly precise and robust. Computational execution was significantly improved with the Picard strategy, and which could be utilized to simulate heterogeneous soil and Newton scheme is superior for homogeneous soil materials.