A Family Of New Globally Convergent Linearization Schemes For Solving Richards' Equation

The Richards' equation models the water flow through porous media in groundwater aquifers and petroleum reservoir simulations. This is a degenerate parabolic differential equation, in which no analytical solution is known. Most numerical methods use implicit time schemes supporting large time steps, but involving near degenerate nonlinear systems. To solve them, we need robust and efficient linearization schemes. Apart from Newton's method, which fails to converge for large time steps and small mesh sizes, recently the globally convergent first order L-scheme was developed, which approximates the derivative by a global upper bound improving Picard's scheme.In this paper, we extend this method to build a family of robust efficient globally convergent first-order linearization schemes by using a sequence of real functions L-n, n >= 0. We solve the space variable by a piecewise linear Galerkin finite element method. The time discretization is based on the Forward-Backward Euler method in the term of hydraulic conductivity K, whereas the remaining terms of the equation dealt with the Backward Euler approximation. We get five schemes improving the convergence of L-schemes and are getting closer to a second-order convergent method. We prove the time iteration convergence in the H-1 (Omega) norm, and we test the schemes on numerical benchmarks to compare them with the L-scheme and Newton's scheme. Among the new schemes, four of them are globally convergent and one scheme is a hybrid case with Newton's scheme. Our results show that among the new schemes, Modified Generalized Linear Scheme (MGLS) obtains the best convergence rates (up to four times that of the L-scheme) using fewer iteration steps and less overall computing time. Finally, our hybrid scheme is robust since it converges when Newton's scheme does not and uses similar computing time, which makes it a good alternative for locally convergent second-order schemes. More methods can be developed using this framework with other L-n sequences and spatial discretizations. (C) 2020 IMACS. Published by Elsevier B.V. All rights reserved.