### Abstract

Language | English |
---|---|

Place of Publication | [Ithaca, NY] |

Number of pages | 104 |

Publication status | Published - 24 Feb 2014 |

Externally published | Yes |

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### Keywords

- Riemann problem
- scalar conservation law
- nonconvex flux
- Euler equations

### Cite this

*The Riemann problem for hyperbolic equations under a nonconvex flux with two inflection points*. [Ithaca, NY].

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**The Riemann problem for hyperbolic equations under a nonconvex flux with two inflection points.** / Fossati, Marco; Quartapelle, Luigi.

Research output: Working paper

TY - UNPB

T1 - The Riemann problem for hyperbolic equations under a nonconvex flux with two inflection points

AU - Fossati, Marco

AU - Quartapelle, Luigi

PY - 2014/2/24

Y1 - 2014/2/24

N2 - This report addresses the solution of Riemann problems for hyperbolic equations when the nonlinear characteristic fields loose their genuine nonlinearity. In this context, exact solvers for nonconvex 1D Riemann problems are developed. First a scalar conservation law for a nonconvex flux with two inflection points is studied. Then the P-system for an isothermal version of the van der Waals gas model is examined in a range of temperatures allowing for a nonconvex pressure function. Eventually the system of the Euler equations of gasdynamics is considered for the polytropic van der Waals gas. In this case, a suitably large specific heat is considered such that the isentropes display a local loss of convexity near the saturation curve and the critical point. Such a nonconvex physical model allows for nonclassical waves to appear as a result of the change of sign of the fundamental derivative of gasdynamics. The solution of the Riemann problem for the considered real gas model reduces to a system of two nonlinear equations for the values of the density on the two sides of the contact discontinuity, much in the same manner of a recently proposed solution method for gases admitting nonlinear wavefields only fully genuine. Solutions containing nonclassical and mixed waves are presented for the three mathematical models. Vacuum formation is described analytically including the presence of composite rarefaction waves, and to the best of authors knowledge, produced numerically here for the first time.

AB - This report addresses the solution of Riemann problems for hyperbolic equations when the nonlinear characteristic fields loose their genuine nonlinearity. In this context, exact solvers for nonconvex 1D Riemann problems are developed. First a scalar conservation law for a nonconvex flux with two inflection points is studied. Then the P-system for an isothermal version of the van der Waals gas model is examined in a range of temperatures allowing for a nonconvex pressure function. Eventually the system of the Euler equations of gasdynamics is considered for the polytropic van der Waals gas. In this case, a suitably large specific heat is considered such that the isentropes display a local loss of convexity near the saturation curve and the critical point. Such a nonconvex physical model allows for nonclassical waves to appear as a result of the change of sign of the fundamental derivative of gasdynamics. The solution of the Riemann problem for the considered real gas model reduces to a system of two nonlinear equations for the values of the density on the two sides of the contact discontinuity, much in the same manner of a recently proposed solution method for gases admitting nonlinear wavefields only fully genuine. Solutions containing nonclassical and mixed waves are presented for the three mathematical models. Vacuum formation is described analytically including the presence of composite rarefaction waves, and to the best of authors knowledge, produced numerically here for the first time.

KW - Riemann problem

KW - scalar conservation law

KW - nonconvex flux

KW - Euler equations

UR - https://arxiv.org/abs/1402.5906

M3 - Working paper

BT - The Riemann problem for hyperbolic equations under a nonconvex flux with two inflection points

CY - [Ithaca, NY]

ER -