An efficient linear scheme to approximate nonlinear diffusion problems
This paper deals with the following nonlinear diffusion problems, including the Stefan problem, the porous medium equation and cross-diffusion systems: \[ \frac{\partial z}{\partial t} = \Delta{\beta} (z) + \mathbf{f}(z)\quad\text{in}\quad \Omega \times (0,\,T), \] \[ \beta (z) = \mathbf{0}\quad\text{on}\quad \partial \Omega \times (0,\,T), \] \[ z( \cdot ,\,0) = z^0 \quad \text{in}\quad \Omega . \] Here, \(\Omega \subset \mathbb{R}^d\,\,(d \in \mathbb{N})\) is a bounded domain with smooth boundary \(\partial \Omega \), \(T\) is a positive constant, \(\beta = (\beta _1 ,\dots,\beta _M )\), \(f = (f_1 ,\dots,f_M ):\) \(\mathbb{R}^M \to \,\, \mathbb{R}^M\) and \(z^0 = (z_1^0 ,\dots,z_M^0 ) \in L^2(\Omega )^M\) are given functions. Numerical scheme to approximate the solution of the given problem is constructed and analyzed. A linear discrete-time scheme was proposed by \textit{A. E. Berger} et al. [RAIRO, Anal. Numér. 13, 297--312 (1979; Zbl 0426.65052)] for degenerate parabolic equations and was extended to cross-diffusion systems by \textit{H. Murakawa} [ESAIM, Math. Model. Numer. Anal. 45, No. 6, 1141--1161 (2011; Zbl 1269.65090)]. There is a constant stability parameter \(\mu \) in the linear scheme. In this paper, a linear discrete-time scheme replacing the constant \(\mu \) with given functions is proposed. After discretizing the scheme in space, an easy-to-implement numerical method for the nonlinear diffusion problems is obtained. Convergence rates of the proposed discrete-time scheme with respect to the time increment are analyzed theoretically. These rates are the same as in the case where \(\mu \) is constant. The paper is organized as follows: Section 1 is an introduction. Notations and definitions are stated in Section 2. The degenerate parabolic equations and the cross-diffusion systems are studied in Sections 3 and 4, respectively. In Section 5, the numerical experiments are carried out in order to investigate the efficiency of the proposed scheme. Finally, some concluding remarks are made in Section 6.
scientific article; zbMATH DE number 6859016
scientific article