On the mathematical analysis and numerical approximation of a system of nonlinear parabolic PDEs (Q732875)

Summary: In this paper we consider a boundary value problem for a system of 2 nonlinear parabolic PDEs e.g. arising in the context of flow and transport in porous media. The flow model is based on the nonlinear Richard's equation problem and is combined with the transport equation through saturation and Darcy's velocity (discharge) terms. The convective terms are approximated by means of the method of characteristics initiated by \textit{P. Pironneau} [Numer. Math. 38, 309--332 (1982; Zbl 0505.76100)] and \textit{J. Douglas jun.} and \textit{T. F. Russel} [SIAM J. Numer. Anal. 19, 871--885 (1982; Zbl 0492.65051)]. The nonlinear terms in Richard's equation are approximated by means of a relaxation scheme applied by \textit{W. Jäger} and \textit{J. Kačur} [RAIRO, Modélisation Math. Anal. Numér. 29, No. 5, 605--627 (1995; Zbl 0837.65103)] and \textit{J. Kačur} [IMA J. Numer. Anal. 19, No. 1, 119--145 (1999; Zbl 0946.65145); SIAM J. Numer. Anal. 36, No. 1, 290--316 (1998; Zbl 0924.65090)]. The convergence of the approximation method is proved.