# Solution of the Cauchy problem for a conservation law with a discontinuous flux function

@article{Gimse1992SolutionOT, title={Solution of the Cauchy problem for a conservation law with a discontinuous flux function}, author={Tore Gimse and Nils Henrik Risebro}, journal={Siam Journal on Mathematical Analysis}, year={1992}, volume={23}, pages={635-648} }

The Cauchy problem is solved for a conservation law arising in oil reservoir simulation where the flux function may depend discontinuously on the space variable. To do this front tracking is used as a method of analysis.

## 174 Citations

### Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition

- Mathematics
- 2017

We study a scalar conservation law whose flux has a single spatial discontinuity. There are many notions of (entropy) solution, the relevant concept being determined by the application. We focus on...

### An Introduction to the Theory of Scalar Conservation Laws with Spatially Discontinuous Flux Functions

- Mathematics, PhysicsApplied Wave Mathematics
- 2009

The aim of these notes is to give a brief introduction to scalar conservation laws with a space-dependent flux function, where the spatial dependence of the flux can have discontinuities.

### Interaction of elementary waves of scalar conservation laws with discontinuous flux function

- Mathematics
- 2006

In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and…

### Conservation laws with discontinuous flux

- MathematicsNetworks Heterog. Media
- 2007

A hyperbolic conservation law with discontinuous flux is considered and a new formulation in terms of Riemann Solvers is provided which provides existence and uniqueness of the corresponding weak entropic solutions.

### The Bifurcation Phenomenon for Scalar Conservation Laws with Discontinuous Flux Functions

- MathematicsActa Applicandae Mathematicae
- 2012

This paper is devoted to study the bifurcation phenomenon for scalar conservation laws with flux functions involving discontinuous coefficients. In order to deal with it, the special Cauchy initial…

### The Bifurcation Phenomenon for Scalar Conservation Laws with Discontinuous Flux Functions

- Mathematics
- 2012

This paper is devoted to study the bifurcation phenomenon for scalar conservation laws with flux functions involving discontinuous coefficients. In order to deal with it, the special Cauchy initial…

### On the convergence of a second order approximation of conservation laws with discontinuous flux

- MathematicsBulletin of the Brazilian Mathematical Society, New Series
- 2016

A second order scheme is constructed for the scalar conservation laws with flux function allowed to be discontinuous in the space variable. The corresponding numerical solutionis shown to converge to…

### On the convergence of a second order approximation of conservation laws with discontinuous flux

- Mathematics
- 2016

A second order scheme is constructed for the scalar conservation laws with flux function allowed to be discontinuous in the space variable. The corresponding numerical solutionis shown to converge to…

### A free boundary problem for scalar conservation laws

- Mathematics
- 1999

We investigate a class of free boundary problems for scalar conservation laws, which is suggested by a model of ion etching. First we give an entropy formulation of the problem; then we prove…

### Vanishing Viscosity and Backward Euler Approximations for Conservation Laws with Discontinuous Flux

- MathematicsSIAM J. Math. Anal.
- 2019

The paper studies the existence of backward Euler approximations, and their convergence to a unique entropy-admissible solution to the Cauchy problem.

## References

SHOWING 1-10 OF 13 REFERENCES

### A NUMERICAL METHOD FOR A SYSTEM OF EQUATIONS MODELLING ONE-DIMENSIONAL THREE-PHASE FLOW IN A POROUS MEDIUM

- Mathematics
- 1989

A numerical method for Riemann problems for a class of equations (system of conservation laws) is presented. Stability and convergence in a case of three-phase flow in porous media is shown and the…

### A NUMERICAL METHOD FOR FIRST ORDER NONLINEAR SCALAR CONSERVATION LAWS IN ONE-DIMENSION

- Mathematics
- 1988

### Global solution of the cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws

- Mathematics
- 1982

### Polygonal approximations of solutions of the initial value problem for a conservation law

- Mathematics
- 1972

### Riemann problems with a discontinuous fiux function

- Proceedings from the Third International Conference on Hyperbolic Problems
- 1990

### Risebro, Riemann problems with a discontinuous fiux function, in "Proceedings from the Third International

- Conference on Hyperbolic Problems," Uppsala,
- 1990

### Analysis of a singular system of conservation laws

- J. Diff. Eqns
- 1986