# Central schemes and contact discontinuities

Alexander Kurganov; Guergana Petrova

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 34, Issue: 6, page 1259-1275
- ISSN: 0764-583X

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topKurganov, Alexander, and Petrova, Guergana. "Central schemes and contact discontinuities." ESAIM: Mathematical Modelling and Numerical Analysis 34.6 (2010): 1259-1275. <http://eudml.org/doc/197505>.

@article{Kurganov2010,

abstract = {
We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes,
proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.],
whose construction is based on the maximal one-sided local speeds of propagation.
We also present a recipe, which helps to improve the resolution of contact waves.
This is achieved by using the partial characteristic decomposition, suggested
by Nessyahu and Tadmor [J. Comput. Phys.87 (1990) 408-463], which is efficiently
applied in the context of the new schemes.
The method is tested on the one-dimensional Euler equations, subject to different
initial data, and the results are compared to the numerical
solutions, computed by other second-order central schemes.
The numerical experiments clearly illustrate the advantages of the
proposed technique.
},

author = {Kurganov, Alexander, Petrova, Guergana},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Euler equations of gas dynamics; partial characteristic
decomposition; fully-discrete and semi-discrete central schemes.; conservation laws; contact discontinuities; high-resolution methods; central schemes; numerical examples; Euler equations},

language = {eng},

month = {3},

number = {6},

pages = {1259-1275},

publisher = {EDP Sciences},

title = {Central schemes and contact discontinuities},

url = {http://eudml.org/doc/197505},

volume = {34},

year = {2010},

}

TY - JOUR

AU - Kurganov, Alexander

AU - Petrova, Guergana

TI - Central schemes and contact discontinuities

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 34

IS - 6

SP - 1259

EP - 1275

AB -
We introduce a family of new second-order Godunov-type central schemes for one-dimensional systems of conservation laws. They are a less dissipative generalization of the central-upwind schemes,
proposed in [A. Kurganov et al., submitted to SIAM J. Sci. Comput.],
whose construction is based on the maximal one-sided local speeds of propagation.
We also present a recipe, which helps to improve the resolution of contact waves.
This is achieved by using the partial characteristic decomposition, suggested
by Nessyahu and Tadmor [J. Comput. Phys.87 (1990) 408-463], which is efficiently
applied in the context of the new schemes.
The method is tested on the one-dimensional Euler equations, subject to different
initial data, and the results are compared to the numerical
solutions, computed by other second-order central schemes.
The numerical experiments clearly illustrate the advantages of the
proposed technique.

LA - eng

KW - Euler equations of gas dynamics; partial characteristic
decomposition; fully-discrete and semi-discrete central schemes.; conservation laws; contact discontinuities; high-resolution methods; central schemes; numerical examples; Euler equations

UR - http://eudml.org/doc/197505

ER -

## References

top- P. Arminjon and M.-C. Viallon, Généralisation du schéma de Nessyahu-Tadmor pour une équation hyperbolique à deux dimensions d'espace. C.R. Acad. Sci. Paris Sér. I320 (1995) 85-88.
- P. Arminjon, M.-C. Viallon and A. Madrane, A finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for conservation laws on unstructured grids. Int. J. Comput. Fluid Dyn.9 (1997) 1-22. Zbl0913.76063
- F. Bianco, G. Puppo and G. Russo, High order central schemes for hyperbolic systems of conservation laws. SIAM J. Sci. Comput.21 (1999) 294-322. Zbl0940.65093
- B. Einfeldt, On Godunov-type methods for gas dynamics. SIAM J. Numer. Anal.25 (1988) 294-318. Zbl0642.76088
- K.O. Friedrichs, Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math.7 (1954) 345-392. Zbl0059.08902
- A. Harten, The artificial compression method for computation of shocks and contact discontinuities. III. Self-adjusting hybrid schemes. Math. Comp.32 (1978) 363-389. Zbl0409.76057
- A. Harten, High resolution schemes for hyperbolic conservation laws. J. Comput. Phys.49 (1983) 357-393. Zbl0565.65050
- A. Harten, B. Engquist, S. Osher and S.R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes III. J. Comput. Phys.71 (1987) 231-303. Zbl0652.65067
- G.-S. Jiang and E. Tadmor, Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM J. Sci. Comput.19 (1998) 1892-1917. Zbl0914.65095
- A. Kurganov, Conservation laws: stability of numerical approximations and nonlinear regularization. Ph.D. thesis, Tel-Aviv University, Israel (1997).
- A. Kurganov and D. Levy, A third-order semi-discrete central scheme for conservation laws and convection-diffusion equations. SIAM J. Sci. Comput. (to appear). Zbl0979.65077
- A. Kurganov and G. Petrova, A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems. Numer. Math. (to appear). Zbl0987.65090
- A. Kurganov, S. Noelle and G. Petrova, Semi-Discrete Central-Upwind Schemes for Hyperbolic Conservation Laws and Hamilton-Jacobi Equations. SIAM J. Sci. Comput. (submitted). Zbl0998.65091
- A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys.160 (2000) 241-282. Zbl0987.65085
- P.D. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math.7 (1954) 159-193. Zbl0055.19404
- B. van Leer, Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov's method. J. Comput. Phys.32 (1979) 101-136.
- D. Levy, G. Puppo and G. Russo, Central WENO schemes for hyperbolic systems of conservation laws. ESAIM: M2AN33 (1999) 547-571. Zbl0938.65110
- D. Levy, G. Puppo and G. Russo, A third order central WENO scheme for 2D conservation laws. Appl. Numer. Math.33 (2000) 407-414. Zbl0965.65106
- D. Levy, G. Puppo and G. Russo, Compact central WENO schemes for multidimensional conservation laws. SIAM J. Sci. Comput.22 (2000) 656-672. Zbl0967.65089
- K.-A. Lie and S. Noelle, Remarks on high-resolution non-oscillatory central schemes for multi-dimensional systems of conservation laws. Part I: An improved quadrature rule for the flux-computation. SIAM J. Sci. Comput. (submitted).
- X.-D. Liu and E. Tadmor, Third order nonoscillatory central scheme for hyperbolic conservation laws. Numer. Math.79 (1998) 397-425. Zbl0906.65093
- H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys.87 (1990) 408-463. Zbl0697.65068
- S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp.50 (1988) 19-51. Zbl0637.65091
- R. Sanders and A. Weiser, A high order staggered grid method for hyperbolic systems of conservation laws in one space dimension. Comput. Methods Appl. Mech. Engrg.75 (1989) 91-107. Zbl0694.65042
- R. Sanders and A. Weiser, High resolution staggered mesh approach for nonlinear hyperbolic systems of conservation laws. J. Comput. Phys.101 (1992) 314-329. Zbl0756.65112
- P. Woodward and P. Colella, The numerical solution of two-dimensional fluid flow with strong shocks. J. Comput. Phys.54 (1988) 115-173. Zbl0573.76057

## Citations in EuDML Documents

top- Smadar Karni, Eduard Kirr, Alexander Kurganov, Guergana Petrova, Compressible two-phase flows by central and upwind schemes
- Smadar Karni, Eduard Kirr, Alexander Kurganov, Guergana Petrova, Compressible two-phase flows by central and upwind schemes
- Alina Chertock, Alexander Kurganov, On a hybrid finite-volume-particle method
- Alexander Kurganov, Doron Levy, Central-upwind schemes for the Saint-Venant system
- Alina Chertock, Alexander Kurganov, On a hybrid finite-volume-particle method
- Alexander Kurganov, Doron Levy, Central-Upwind Schemes for the Saint-Venant System

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