Conservation law with the flux function discontinuous in the space variable. II: Convex-concave type fluxes and generalized entropy solutions
type: Item
instance of: scholarly article
title: Conservation law with the flux function discontinuous in the space variable. II: Convex-concave type fluxes and generalized entropy solutions
zbMATH Open document ID: 1131.65071
author: Adimurthi
author: Siddhartha Mishra
author: G. D. Veerappa Gowda
published in: Journal of Computational and Applied Mathematics
publication date: 2007-05-11T00:00:00Z
review text: Following one-dimensional nonlinear Cauchy problem is investigated: \[ u_{t}+(H(x)f(u)+(1-H(x))g(u))_{x}=0; \quad u(x,0)=u_{0}(x) \] where \(f\) and \(g\) are Lipschitz continuous and \(H\) is the Heaviside function. Such an equation arises, e.g., in the two phase flows in petroleum reservoirs. The authors develop the concept of generalized entropy solutions for such equations by replacing the Rankine-Hugoniot condition by a generalized Rankine-Hugoniot condition. In a recent work [J. Hyperbolic Differ. Equ. 2, No.~4, 783--837 (2005; Zbl 1093.35045)] the authors have proposed a new entropy framework for the equation of the above type -- the so called ``optimal entropy solutions''. Under the assumption that \( f,g \) have the same type of convexity they have shown the existence of infinitely many stable solutions. As for the weak solutions, they may not exist. A Godunov type of convergent finite difference scheme allows to compute a numerical solution and in the same time serves to prove existence theorems. The last section is devoted to numerical experiments. [For part I see \textit{Adimurthi} and \textit{G. D. Veerappa Gowda}, J. Math. Kyoto Univ. 43, No.~1, 27--70 (2003; Zbl 1063.35114).]
Mathematics Subject Classification ID: 65M06
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35F25
Mathematics Subject Classification ID: 35L65
zbMATH DE Number: 5151704
zbMATH Keywords: conservation law
zbMATH Keywords: Godunov finite difference method
zbMATH Keywords: entropy solutions
zbMATH Keywords: Rankine-Hugoniot condition convergence
zbMATH Keywords: nonlinear Cauchy problem
zbMATH Keywords: two phase flows
zbMATH Keywords: optimal entropy solutions
zbMATH Keywords: numerical experiments
MaRDI profile type: Publication
full work available at URL: j.cam.2006.04.009
OpenAlex ID: W2059991999
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DOI: 10.1016/J.CAM.2006.04.009
description: scientific article; zbMATH DE number 5151704