# A Comparative Study of the Transformed Methods for Solving Richards' Equation

*by*M. Rashidul Islam,

**Abstract:**The Richards' equation represents the movement of water in both saturated and unsaturated porous medium. The numerical solution to this equation is difficult because the algebraic equations derived from numerical discretization are non-linear and requires an iterative technique for the solution. Transformation methods applied to the Richards' equation reduce the non-linearity and numerical solutions to the transformed equations become less tedious and offer a relatively faster convergence. In this paper, the mixed-form Richards' equation is first solved by finite element method coupled with mass-lumping using linear weighting function. Picard iteration technique is adopted for the non-linear iterations. The computed results are compared with those of Celia et al. (1990). Two transformation methods, logarithmic and hyperbolic sine, are applied to matric potential to reduce the non-linearity of the equation. The transformed equations are then solved by the same method adopted for the original equation. Each of the transformation techniques results in a significant reduction in the number of iterations required for the solution to converge. The solution converges rapidly for the transformed equation since the transformation reduces the non-linearity in the equation. The number of iterations required to converge the transformed equations are approximately 25% of those required by the untransformed equations.

**Subject Headings:**Nonlinear analysis | Finite element method | Comparative studies | Water management | Porous media | Convergence (mathematics) | Coupling

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