An efficient linear scheme to approximate nonlinear diffusion problems
type: Item
instance of: scholarly article
title: An efficient linear scheme to approximate nonlinear diffusion problems
zbMATH Open document ID: 1400.35163
author: Hideki Murakawa
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 2018-04-12T00:00:00Z
review text: 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. Fina…
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 80A22
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65F05
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 76S05
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 35Q79
zbMATH DE Number: 6859016
zbMATH Keywords: Stefan problem
zbMATH Keywords: porous medium equation
zbMATH Keywords: cross-diffusion system
zbMATH Keywords: linear scheme
zbMATH Keywords: error estimate
zbMATH Keywords: numerical method
reviewed by: Temur Jangveladze
MaRDI profile type: Publication
full work available at URL: s13160-017-0279-3
OpenAlex ID: W2767487626
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DOI: 10.1007/S13160-017-0279-3
description: scientific article; zbMATH DE number 6859016