Riemann solution for one dimensional non-ideal isentropic magnetogasdynamics. (English) Zbl 1342.35258
Summary: In the present paper, we study the Riemann problem for quasilinear hyperbolic system of partial differential equations governing the one dimensional non-ideal isentropic magnetogasdynamics with transverse magnetic field. We discuss the properties of rarefaction waves, shocks and contact discontinuities. Differently from single equation methods rooted in the ideal gasdynamics, the new approach is based on the system of two nonlinear algebraic equations imposing the equality of total pressure and velocity, assuming as unknowns the two values of densities, on both sides of the contact discontinuity. Newton iterative method is used to obtain densities. The resulting exact solver is implemented with the examples of general applicability of the proposed approach.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35L60 | First-order nonlinear hyperbolic equations |
| 76N15 | Gas dynamics (general theory) |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
| 65H10 | Numerical computation of solutions to systems of equations |
| 76L05 | Shock waves and blast waves in fluid mechanics |
Keywords:
Riemann problem; magnetogasdynamics; non-ideal gas; shock wave; rarefaction wave; contact discontinuityReferences:
| [1] | Arora R, Tomar A, Singh VP (2012) Similarity solutions for strong shocks in a non-ideal gas. Math Model Anal 17(3):351-365 · Zbl 1259.35139 · doi:10.3846/13926292.2012.685957 |
| [2] | Cabannes H (1970) Theoretical magnetofluid dynamics. In: Applied mathematics and mechanics, vol 13. Academic Press, New York · Zbl 1271.76388 |
| [3] | Courant R, Friedrichs KO (1999) Supersonic flow and shock waves. Interscience, New York · Zbl 0041.11302 |
| [4] | Glimm J (1965) Solutions in the large for nonlinear hyperbolic systems of equations. Commun Pure Appl Math 18:697-715 · Zbl 0141.28902 · doi:10.1002/cpa.3160180408 |
| [5] | Gundersen R (1964) Linearized analysis of one-dimensional magnetohydrodynamic flows. Springer, Berlin · Zbl 0123.22301 · doi:10.1007/978-3-642-46005-0 |
| [6] | Kuila S, Raja Sekhar T (2014) Riemann solution for ideal isentropic magnetogasdynamics. Meccanica 49(10):2453-2465 · Zbl 1299.76301 |
| [7] | Landau LD, Lifshitz EM (2005) Electrodynamics of continuous media, 2nd edn. Elsevier, Amsterdam · Zbl 0081.08803 |
| [8] | Lax PD (1957) Hyperbolic systems of conservation laws, II. Commun Pure Appl Math 10:537-566 · Zbl 0081.08803 · doi:10.1002/cpa.3160100406 |
| [9] | Liu Y, Sun W (2013) Riemann problem and wave interactions in magnetogasdynamics. J Math Anal Appl 397(2):454-466 · Zbl 1256.35087 · doi:10.1016/j.jmaa.2012.07.064 |
| [10] | Myong RS, Roe PL (1997a) Shock waves and rarefaction waves in magnetohydrodynamics part 1. A model system. J Plasma Phys 58:485-519 · Zbl 1183.35230 |
| [11] | Myong RS, Roe PL (1997b) Shock waves and rarefaction waves in magnetohydrodynamics part 2. The MHD system. J Plasma Phys 58:521-552 |
| [12] | Quartapelle L, Castelletti L, Guardone A, Quaranta G (2003) Solution of the Riemann problem of classical gasdynamics. J Comput Phys 190:118-140 · Zbl 1236.76054 · doi:10.1016/S0021-9991(03)00267-5 |
| [13] | Raja Sekhar T, Sharma VD (2010) Riemann problem and elementary wave interactions in isentropic magnetogasdynamics. Nonlinear Anal Real World Appl 11(2):619-636 · Zbl 1183.35230 · doi:10.1016/j.nonrwa.2008.10.036 |
| [14] | Shen C (2011) The limits of Riemann solutions to the isentropic magnetogasdynamics. Appl Math Lett 24(7):1124-1129 · Zbl 1215.35105 · doi:10.1016/j.aml.2011.01.038 |
| [15] | Singh LP, Husain A, Singh M (2011) A self-similar solution of exponential shock waves in non-ideal magnetogasdynamics. Meccanica 46(2):437-445 · Zbl 1271.76388 · doi:10.1007/s11012-010-9325-9 |
| [16] | Smoller J (1994) Shock waves and reaction-diffusion equations. Springer, New York · Zbl 0807.35002 · doi:10.1007/978-1-4612-0873-0 |
| [17] | Toro EF (1997) Riemann solvers and numerical methods for fluid dynamics, 2nd edn. Springer, Berlin · Zbl 0888.76001 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
