An adaptive strategy for discontinuous Galerkin simulations of Richards’ equation: Application to multi-materials dam wetting

Abstract

Numerical solution of Richards' equation remains challenging to get robust, accurate and cost-effective results, particularly for moving sharp wetting fronts. An adaptive strategy for both space and time is proposed to deal with 2D sharp wetting fronts associated with varying and possibly vanishing diffusivity caused by nonlinearity, heterogeneity and anisotropy. Adaptive time stepping makes nonlinear convergence reliable and backward difference formula provides high-order time scheme. Adaptive mesh refinement tracks wetting fronts with an a posteriori error indicator. The novelty of this paper consists in using this technique in combination with a weighted discontinuous Galerkin framework to better approximate steep wetting fronts by a discontinuity. The potential of the overall approach is shown through various examples including analytical and laboratory benchmarks and simulation of full-scale multi-materials dam wetting experiment.

A Richards’ equation-based model for wave-resolving simulation of variably-saturated beach groundwater flow dynamics

“…Then, the seepage boundary condition may be interpreted as a nonlinear Robin boundary condition. In this study, it is treated inside the nonlinear iterative process according to the previous solution guess at a local level of the numerical scheme, see [45] for further insights.…”

“…To solve efficiently Richards' equation, the work presented here uses a strategy based on a Discontinuous Galerkin method in combination with adaptive mesh refinement. This strategy has been introduced in Clément et al [45] and successfully applied to a multi-materials dam wetting problem. Extensive test-cases are also available in [46].…”

Forward and inverse modeling of water flow in unsaturated soils with discontinuous hydraulic conductivities using physics-informed neural networks with domain decomposition

“…For example, when simulating water flow under infiltration based on the RRE using traditional numerical methods, the spatial mesh must be very small near the soil surface to obtain reliable solutions, which is computationally very expensive or intractable for a threedimensional watershed scale. Although there is progress in numerical methods other than PINNs, including a discontinuous Galerkin method with adaptive spatial and temporal meshes (Clément et al, 2021), it is worthwhile to seek the potential of PINNs to solve the RRE.…”

Optimizing Water Consumption in Richards’ Equation Framework with Step-Wise Root Water Uptake: A Simplified Model

“…such problem; it is worth citing, for instance, studies on time-stepping techniques (mainly low-order implicit methods, as implicit Euler, see Pop 2002;Kavetski et al 2001;Keita et al 2021), or different spatial discretization techniques such as mixed finite element or finite volume methods-endowed also with rigorous error estimates- (Arbogast et al 1993(Arbogast et al , 1996Eymard et al 1999Eymard et al , 2006Manzini and Ferraris 2004;Radu et al 2004Radu et al , 2008, discontinuous Galerkin (e.g., Clément et al 2021;Li et al 2007), gradient discretization schemes (Eymard et al 2014), andvirtual element methods (da Veiga et al 2021). Equally challenging is the problem of solving the nonlinear system arising from the time integration: to this purpose, many schemes have been proposed, as L-scheme (List and Radu 2016;Mitra and Pop 2018), Newton and nested-Newton (Bergamaschi and Putti 1999;Casulli and Zanolli 2010), modified Picard (Celia et al 1990), and a combination of Newton and Picard methods (Lehmann and Ackerer 1998).…”