For the nonlinear Richards equation as an unsaturated flow through heterogeneous media, we build a new coarse-scale approximation algorithm utilizing numerical homogenization. This approach follows deep neural networks (DNNs) to quickly and frequently calculate macroscopic parameters. More specifically, train with training set consisting of stochastic permeability realizations corresponding com...

We extend our recent work on higher-order time integration of Richards' equation to layered heterogeneous porous media, using a diierential-algebraic-equation-based method of lines (DAE/MOL) approach. We show that the DAE/MOL approach is robust and eecient compared to standard low-order time integration methods for heterogeneous media. We also show the advantage of using an integral representat...

We introduce a new model of traffic flow with phase transitions. The model is obtained by coupling together the classical Lighthill-Whitham-Richards (LWR) equation with the 2 × 2 system described in [A. Aw and M. Rascle, Resurrection of “second order” models of traffic flow, SIAM J. Appl. Math., 60 (2000), pp. 916–938]. We describe the solutions of the Riemann problem, and we compare the result...

The nonlinear solvers in numerical solution of water flow in variably saturated soils are prone to convergence difficulties. Many aspects can give rise to such difficulties, like very dry initial conditions, a steep pressure gradient and great variation of hydraulic conductivity occur across the wetting front during the infiltration of water.  So, the averaging method applied to compute hydraul...

Water flow into unsaturated porous media is governed by the Richards’ partial differential equation expressing the mass conservation law and the Darcy ́s law. This equation may be written in three standard forms: the “h-based” form, the “θ-based” form, and the “mixed” form. Numerical aproximations based on different forms of this one-dimensional nonlinear equation can lead to significantly diffe...