Estimation of optimal penalization parameters and convergence of the non-linear solver for a Richards' equation-based model for variably-saturated groundwater flows based on an IIPG Discontinuous Galerkin method
Résumé
In this article, we study the convergence of an IIPG (Incomplete Interior Penalty Galerkin) Discontinuous Galerkin numerical method for the Richards equation. The Richards equation is a degenerate parabolic nonlinear equation for modeling flows in porous media with variable saturation. The numerical solution of this equation is known to be difficult to calculate numerically, due to the abrupt displacement of the wetting front, mainly as a result of highly non-linear hydraulic properties. As time scales are slow, implicit numerical methods are required and the convergence of nonlinear solvers is very sensitive. We propose an original method to ensure convergence of the numerical solution to the exact Richards solution, using a technique of auto-calibration of the penalty parameters derived from the Galerkin Discontinuous method. The method is constructed using non-linear 1D and 2D general elliptic problems. We show that the numerical solution converges toward the unique solution of the continuous problem under certain conditions on the penalty parameters. Then, we numerically demonstrate the efficiency and robustness of the method through test cases with analytical solutions, laboratory test cases, and large-scale simulations.
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