Conservative temporal and spatial adaptive methods for groundwater flow

Variably saturated groundwater flow problems are often modeled by Richards equation [14][19], a nonlinear partial differential equation. Efficient and robust numerical approximations of Richards model continue to be challenging for most common problems due to the nonlinearities of the model and the nonsmooth properties of its constitutive relations for certain physical parameters. These problems can lead to a series of difficulties including loss of mass, poorly resolved fronts and failure for nonlinear and iterative linear solvers [11] [16] [26] [34] [43] [50] . Standard methods that use uniform temporal and spatial discretization for these problems are often inefficient and have given way to dynamic methods that adapt in time and space. While the advantages of each component of such adaption strategies have been demonstrated, the joint use of spatially and temporally adaptive methods for solving Richards' equation has received little or no attention [33]. This work has two parts; joint adaption for Richards' equation and mass conservative issues associated with spatial discretizations. For the joint adaption, we propose a method for solving Richards' equation which is adaptive in both space and time. Next we present conservation schemes that address mass conservation issues associated with grid coarsening in spatial adaption. The motivation for this work is the ADaptive Hydrology Model(ADH)[47], being developed by the U.S. Army corps of Engineers, for solving surface and ground water flow problems. ADH couples 3D unsaturated ground water modeling to 2D shallow water modeling at the surface. It advances in time implicitly, solving the nonlinear equations with an inexact Newton method with a two level domain decomposition preconditioner. We provide promising numerical results using a 1-D simulation tool structured similarly to ADH that incorporates a joint spatial and temporal adaption scheme with a choice of three methods for mass conservation.