Abstract
A short review on central schemes for the numerical solution of hyperbolic systems of conservation laws is given. The main focus of the talk concerns the construction of high order central schemes in one and two space dimensions, and the extension of central schemes to systems with source term. The treatment of stiff source and the construction of well-balanced central scheme will be addressed.
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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.
Arminjon P., StanescuD.Viallon M.-C., A Two-Dimensional Finite Volume Extension of the Lax-Friedrichs and Nessyahu-Tadmor Schemes for Compressible Flow, Proc. 6th. Int. Symp. on CFD, Lake Tahoe, 1995, M. Hafez and K. Oshima, editors, Vol. IV, 7–14.
Arminjon P., Viallon M.-C. and Madrane A., 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.
F. Bianco, G. Puppo, G. Russo, High Order Central Schemes for Hyperbolic Systems of Conservation Laws, SIAM J. Sci. Comp., 21 (1999), 294–322.
R. E. Caflisch, S. Jin and G. Russo, Uniformly accurate schemes for hyperbolic systems with relaxation, SIAM J. Numer. Anal., 34 (1997), 246–281.
R. Gatignol, Théorie cinétique des gaz à répartition discréte de vitesses. Lectures Notes in Physics, 36 (1975) Springer-Verlag.
L. Gosse, A Well-Balanced Flux-Vector Splitting Scheme Designed for Hyperbolic Systems of Conservation Laws with Source Terms, Comput. Math. Appl. 39 (2000), 135–159.
A. Harten, B. Engquist, S. Osher, S.R. Chakravarthy, Uniformly High Order Accurate Essentially Nonoscillatory Schemes. III, J. Comput. Phys., 71 (1987), 231–303.
Jiang G.-S., Levy D., Lin C.-T., Osher S., Tadmor E. High-Resolution Non-Oscillatory Central Schemes with Non-Staggered Grids for Hyperbolic Conservation Laws, SIAM J. Numer. Anal., 35 (1998), 2147–2168.
G.-S. Jiang, E. Tadmor, Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws, SIAM J. Sci. Comp., 19 (1998), 1892–1917.
R. Kupferman, E. Tadmor, A fast,high resolution, second-order central scheme for incompressible flow, Proc.Natl.Acad.Sci.USA 94 (1997), 4848–4852.
A. Kurganov, D. Levy, A Third-Order Semi-Discrete Central Scheme for Conservation Laws and Convection-Diffusion Equations,SIAM J. Sci. Comput. 22 (2000), 1461–1488.
A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys. 160 (2000), 241–282.
R.J. LeVequeBalancing source terms and flux gradients in high-resolution Godunov methods: the quasi-steady wave-propagation algorithm, J. Comput. Phys.146346–365 (1998).
D. Levy, G. Puppo, G. Russo, Central WENO Schemes for Hyperbolic Systems of Conservation Laws, M2AN, 33 (1999), 547–571.
D. Levy, G. Puppo, G. Russo, Compact Central WENO Schemes for Multidimensional Conservation Laws, SIAM J. Sci. Comput. 22 (2000), 656–672.
D. Levy, E. TadmorNon-oscillatory central schemes for the incompressible Euler equations, Mathematical Research Letters, 4 (1997), 321–340.
S. F. Liotta, V. Romano and G. Russo, Central schemes for balance laws of relaxation type, SIAM J. Numer. Anal. 38 (2000), 1337–1356.
Liu X.-D., Osher S., Chan T., Weighted Essentially Non-oscillatory Schemes, J. Comput. Phys., 115 (1994), 200–212.
Xu-Dong Liu, S. Osher, Convex ENO high order multi-dimensional schemes without field by field decomposition or staggered grids, J. Comput. Phys. 142 (1998), 304–330.
X.-D. Liu, E. Tadmor, Third Order Nonoscillatory Central Scheme for Hyperbolic Conservation Laws, Numer. Math., 79 (1998), 397–425.
I. M¨¹ller and T. Ruggeri, Rational extended thermodynamics, (1998) Springer-Verlag, Berlin.
H. Nessyahu, E. Tadmor, Non-oscillatory Central Differencing for Hyperbolic Conservation Laws, J. Comput. Phys., 87 (1990), 408–463.
G. Russo, Well Balanced central schemes, in preparation.
R. Sanders and W. Weiser, A High Resolution Staggered Mesh Approach for Nonlinear Hyperbolic Systems of Conservation Laws, J. Comput. Phys., 10 (1992), 314–329.
Shu C.-W., Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, Lecture Notes in Mathematics (editor: A. Quarteroni), Springer, Berlin, 1998.
J.J. Stoker, Water waves, (1957) Interscience Publishers, New York.
Zennaro M., Natural Continuous Extensions of Runge-Kutta Methods, Math. Comp., 46 (1986), 119–133.
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Russo, G. (2001). Central Schemes for Balance Laws. In: Freistühler, H., Warnecke, G. (eds) Hyperbolic Problems: Theory, Numerics, Applications. ISNM International Series of Numerical Mathematics, vol 141. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8372-6_35
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DOI: https://doi.org/10.1007/978-3-0348-8372-6_35
Publisher Name: Birkhäuser, Basel
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