Riemann problem for one-dimensional system of conservation laws of mass, momentum and energy in zero-pressure gas dynamics

Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation lawswhere σ(u) = u3 − u. This system is hyperbolic at (u, v) unless . We find that the Riemann problem has a unique solution for all data in the hyperbolic regions, except for a range of data in the same phase (i.e. on the same side of the nonhyperbolic strip). In the nonunique cases, there are exactly two admissible solutions. The analysis is based upon a formula describing all saddle-to-saddle heteroclinic orbits for a family of cubic vector fields in the plane.

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The Riemann problem for a resonant nonlinear system of conservation laws with Dirac-measure solutions

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500027165 ◽

1998 ◽

Vol 128 (1) ◽

pp. 81-94 ◽

Cited By ~ 9

Author(s):

Jiaxin Hu

Keyword(s):

Nonlinear System ◽

Conservation Laws ◽

Riemann Problem ◽

Additional Condition ◽

Dirac Measure ◽

Viscosity Approximation ◽

Riemann Solution ◽

System Of Conservation Laws ◽

Genuine Nonlinearity ◽

Contact Wave

The Riemann problem for a resonant nonlinear system of conservation laws is considered here. The Riemann solution was constructed by employing the viscosity approximation approach. One kind of new discontinuity, which is called the Dirac-contact wave, appeared in the Riemann solution. Because the strict hyperbolicity as well as the genuine nonlinearity of the system considered failed, the solution we obtained in this paper is not unique for some initial data. An additional condition was explored to guarantee the uniqueness of the Riemann problem.

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On invariant finite-difference schemes for equations of one-dimensional flows of a polytropic gas for problems with spatial symmetries

Keldysh Institute Preprints ◽

10.20948/prepr-2021-92 ◽

2021 ◽

pp. 1-34

Author(s):

Aleksander Alekseevich Russkov ◽

Evgeny Igorevich Kaptsov

Keyword(s):

Conservation Laws ◽

Gas Dynamics ◽

State Equation ◽

Difference Schemes ◽

Finite Difference Schemes ◽

Local Conservation ◽

Gas Dynamics Equations ◽

One Dimensional ◽

Polytropic Gas ◽

Invariant Properties

One-dimensional polytropic gas dynamics equations for plane, radially symmetric, and spherically symmetric flows are considered. Invariant properties of equations are discussed, local conservation laws are derived. Additional conservation laws are written, which take place only in case of special values of adiabatic exponent. Classical difference scheme of Samarsky-Popov for gas dynamics has all difference analogs of conservation laws, except for additional ones. In difference schemes additional conservative laws take place in case of special state equation approximation. Scheme of Samarsky-Popov with special state equation was initially suggested by V.A. Korobitsyn. He described it as ‘thermodynamically consistend’ In current paper group properties, and conservation laws of thermodynamically consistent schemes are discussed, and numerical implementation for plane, cylinder, and spherical flows is perfomed.

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