Paper Info
Title
Convergence of Upwind Finite Difference Schemes for a Scalar Conservation Law with Indefinite Discontinuities in the Flux Function
Abstract
We consider the scalar conservation law with flux function discontinuous in the space variable, i.e., \begin{eqnarray} \label{eq1} u_t+(H(x)f(u)+(1-H(x))g(u))_{x} &=& 0 \quad \mbox{in } \R \times \R_{+}, \nonumber \\ u(0, x) &=& u_{0}(x) \quad \mbox{in } \R, \label{0.1} \end{eqnarray} where $H$ is the Heaviside function and $f$ and $g$ are smooth with the assumptions that either $f$ is convex and $g$ is concave or $f$ is concave and $g$ is convex. The existence of a weak solution of (\ref{eq1}) is proved by showing that upwind finite difference schemes of Godunov and Enquist--Osher type converge to a weak solution. Uniqueness follows from a Kruzkhov-type entropy condition. We also provide explicit solutions to the Riemann problem for (\ref{eq1}). At the level of numerics, we give easy-to-implement numerical schemes of Godunov and Enquist--Osher type. The central feature of this paper is the modification of the singular mapping technique (the main analytical tool for these types of equations) which allows us to show that the numerical schemes converge. Equations of type (\ref{eq1}) with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux: \begin{equation} \label{eq2} \begin{array}{r@{\;}l} u_t + (k (x) f (u))_x &= 0, \\ u (0, x) &= u_0 (x), \end{array} \end{equation} with $f$ convex and $k$ of indefinite sign.
Year
DOI
Venue
2005
10.1137/030602745
SIAM J. Numerical Analysis
Keywords
Field
DocType
indefinite discontinuities,easy-to-implement numerical scheme,weak solution,numerical schemes converge,following scalar conservation law,osher type converge,heaviside function,flux function discontinuous,upwind finite difference schemes,flux function,osher type,discontinuous flux,explicit solution,scalar conservation law,finite differences,conservation laws,mathematics
Uniqueness,Mathematical optimization,Mathematical analysis,Weak solution,Regular polygon,Godunov's scheme,Upwind scheme,Conservation law,Mathematics,Riemann problem,Heaviside step function
Journal
Volume
Issue
ISSN
43
2
0036-1429
Citations 
PageRank 
References 
5
0.79
0
Authors
1
Name
Order
Citations
PageRank
Siddhartha Mishra117021.36