Abstract
Several models in mathematical physics are described by quasilinear hyperbolic systems with source term, which in several cases may become stiff. Here a suitable central numerical scheme for such problems is developed and application to shallow water equations, Broadwell model and Extended Thermodynamics are mentioned.
The numerical methods are a generalization of the Nessyahu-Tadmor scheme to the non-homogeneous case. We propose two ways for treating the production term. The first is obtained by including the cell averages of the productions, while the second family of schemes is obtained by a splitting strategy.
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References
A. M. Anile and O. Muscato, Improved hydrodynamical model for carrier transport in semiconductors, Phys. Rev. B, 51 (1995), 16728–16740.
A. M. Anile and S. Pennisi, Thermodynamic derivation of the hydrodynamical model for charge transport in semiconductors, Phys. Rev. B, 46 (1992), 13186–13193.
A. M. Anile, V. Romano, and G. Russo, Extended Hydrodynamical Model of Carrier Transport in Semiconductors, submitted to SIAM J. of Appl. Math.
J. D. Au, Nonlinear Closure in Molecular Extended Thermodynamics with Application to the Shock Wave Structure Solution, in Supplemento ai Rendiconti del Circolo Matematico di Palermo 45, Proceedings International Conference on Wave and Stability, A. Greco and S. Rionero ed.s, Palermo, 1995.
R. E. Caflisch, S. Jin and G. Russo, Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation, SIAM J. Num. Anal., 34 (1997), 246–281.
G. Q. Chen, C. D. Levermore and T. P. Liu, Hyperbolic Conservation Laws with Stiff Relaxation Terms and Entropy, Comm. Pure Appl. Math., Vol. XLVII, (1994), 787–830.
G. Erbes, A high-resolution Lax-Friedrichs scheme for hyperbolic conservation laws with source terms. Application to the shallow water equations, preprint.
R. Gatignol, Lectures Notes in Physics, vol. 36 Springer-Verlag, 1975.
A. Harten and S. Osher, Uniformly High-Order Accurate Nonoscillatory Schemes. I, SIAM J. Numer. Anal. 24 (1987), 279–309.
I-Shi Liu, Method of Lagrange Multipliers for Exploitation of the entropy principle, Arch. Rat. Mech. Anal., 46 (1972), 131–148.
S. Jin, Runge-Kutta Methods for Hyperbolic Conservation Laws with Stiff Relaxation Terms, J. Comput. Phys., 122 (1995), 51–67.
D. Jou, J. Casas-Vazquez and G. Lebon, Extended irreversible thermodynamics, Springer-Verlag, Berlin, 1993.
S. F. Liotta, V. Romano, and G. Russo, Central Methods for Systems of Balance Laws, with Application to Continuous Mechanics, in preparation.
D. Mihalas & B. Mihalas, Foundations of Radiation Hydrodynamics, Oxford UP, 1984.
I. Müller e T. Ruggeri, Extended thermodynamics, Springer-Verlag, Berlin, 1993.
H. Nessyahu and E. Tadmor, Non-Oscillatory Central Differencing for Hyperbolic Conservation Laws, J. Comput. Phys., 87 (1990) 408–448.
L. Pareschi, M. Ronconi, G. Russo, Uniformly accurate central Schemes for Hyperbolic Systems with Relaxation, in preparation.
R. B. Pember, Numerical Methods for Hyperbolic Conservation Laws with Stiff Relaxation I. Spurious Solutions, SIAM J. Appl. Math., 53 (1993), 1293–1330.
R. Sanders and W. Weiser, High Resolution Staggered Mesh Approach for Nonlinear Hyperbolic Systems of Conservation Laws, J. Comput. Phys., 101 (1992), 314–329.
J. J. Stoker, Water Waves, Intersciennce Publishers, New York, 1974.
W. Vincenti and C. Kruger, Introduction to Physical Gas Dynamics, Krieger Publishing Company, Malabar, Florida, 1986.
G. B. Whitham, Linear and non linear waves, Wiley, New York, 1974.
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Liotta, S.F., Romano, V., Russo, G. (1999). Central Schemes for Systems of Balance Laws. In: Jeltsch, R., Fey, M. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 130. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8724-3_16
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DOI: https://doi.org/10.1007/978-3-0348-8724-3_16
Publisher Name: Birkhäuser, Basel
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Online ISBN: 978-3-0348-8724-3
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