Grant: Discovery Grants Program - Individual

The aim of the proposed research program is to develop new numerical methods for modelling infiltration processes in heterogeneous soils and their interactions with surface flows. We will propose robust computational techniques for the simulation of unstable infiltration in heterogeneous soils, develop efficient numerical models for coupled surface-subsurface flows, propose error estimators and develop new space-time techniques to improve the accuracy and efficiency of the proposed numerical models. We will use the new continuum model that we have recently developed which accurately describes the formation of gravity fingers during infiltration in soils. Robust and efficient two-dimensional and three-dimensional numerical approximation methods will be developed for a new generation of multiphase-flow models in porous media known as phase-field models. Phase-field models are based on the fact that the non-homogeneity of the system and the macroscopic interfaces should be considered for computing the energy of the system. The sharp interfaces separating different phases in the system should be replaced by diffusive interfaces by the introduction of a diffusive order parameter or phase-field parameter that varies smoothly over thin thickness regions. The spatiotemporal evolution in these regions can be described by a set of coupled partial differential equations and solved numerically using suitable discretization techniques. The time-evolution of the phase-field parameter is derived from the variational principle by minimizing the free energy expression of the system. High-order accurate discretizations will be developed and a suitable temporal integration will be used to deal with the issues of the instability associated with the sources of stiffness in the numerical solutions. The flux terms of the governing equations of the coupled system of surface-subsurface flows can be split into a hyperbolic part and an elliptic part. Non-oscillatory high-order continuous/discontinuous reconstructions of the solutions will be developed in order to design stable and accurate numerical schemes. A suitable temporal scheme is required to deal with the issues of the instability associated with the sources of stiffness in the numerical solutions of the system. The semi-discrete form of the system can be split into linear stiff terms and residual nonlinear non-stiff terms, and implicit time integration methods will be used for stiff terms and the explicit temporal schemes can be used to integrate non-stiff terms. In our approach, we will use errors estimators, mesh adaptation, and propose new technique to accurately capture the propagation of the wetting fronts. The efficiency, convergence and accuracy of the proposed numerical models will be examined by several numerical tests. The spatial convergence rate of the proposed schemes will be determined numerically and the impact of the semi-implicit approach on the temporal order of these methods will be studied.