The finite volume method for Richards equation (Q1977971)

The authors prove the convergence of a finite volume scheme for the discretization of an elliptic-parabolic problem, namely Richards equation \(\beta (P)_t- \text{div} (K(\beta (P))\nabla (P+ z))= 0\) with Dirichlet boundary conditions and an initial condition. This is done by means of a priori estimates in \(L^2\), and by the use of Kolmogorov's theorem on relative compactness of subsets of \(L^2\). The problem is considered in the form \(c(u)_t =\Delta u+ \text{div} (K(c(u)) \nabla z)\) in \(Q_T= \Omega \times (0,T)\), \(u =u^D\) in \(\partial \Omega \times (0,T)\), \(c(u(x,0))= c(u_0 (x))\) for all \(x \in \Omega\), where \(\Omega \) is an open bounded domain of \({\mathcal R}^N \) with smooth boundary \(\partial \Omega \), and \(T\) is a positive constant. The space coordinates are denoted by \(x =(x_1,x_2, \ldots,x_{N- 1},z)\), and it is assumed that \(c\) is a Lipschitz continuous nondecreasing function with Lipschitz constant \(L_c \), \(K\) is a nondecreasing Lipschitz continuous function with Lipschitz constant \(L_K \), and initial condition \(u_0 \) and boundary data \(u^D \) satisfy the hypotheses \(u_0 \in L^\infty (\Omega)\), \(u^D\) is Lipschitz continuous on \(\bar \Omega \) with Lipschitz constant \(L_D\). The equation changes type in \(\Omega \times {\mathcal R}^+ \): it is elliptic in the regions where \(c(u)\) is constant, and parabolic elsewhere. Since the solution is not expected to be smooth, a weak solution of the problem is defined. It follows from a result by \textit{F. Otto} [J. Differ. Equations 131, No. 1, 20-38, Art. No. 0155 (1996; Zbl 0862.35078)] that the problem has at most one weak solution. The authors introduce a finite volume scheme and define the approximate problems \((P_{h,k})\). The upwinding is used in the discretization of convection and permits one to avoid numerical oscillations that are present otherwise. The existence and uniqueness of the solution \(u_{h,k}\) for problem \((P_{h,k})\) are proved. A priori estimates are derived, and from these estimates the authors derive the existence of a subsequence of \(\{u_{h,k}\}\) which converges to a function \(u \in L^2 (0,T;H^1 (\Omega))\) weakly in \(L^2 (Q_T)\), such that \(\{c(u_{h,k})\}\) converges to a function \(\chi\) strongly in \(L^2 (Q_T)\). Then, it is proved that \(\chi= c(u)\), and that \(u\) is the unique weak solution of the above continuous problem. Finally, numerical results for two numerical tests are presented.