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A high-order ENO conservative Lagrangian type scheme for the compressible Euler equations. (English) Zbl 1126.76035
Summary: We develop a class of Lagrangian type schemes for solving Euler equations of compressible gas dynamics both in Cartesian and cylindrical coordinates. The schemes are based on high-order essentially non-oscillatory (ENO) reconstruction. They are conservative for density, momentum and total energy, can maintain formal high-order accuracy both in space and time, can achieve at least uniformly second-order accuracy with moving and distorted Lagrangian meshes, are essentially non-oscillatory, and have no parameters to be tuned for individual test cases. One- and two-dimensional numerical examples in Cartesian and cylindrical coordinates are presented to demonstrate the performance of the schemes in terms of accuracy, resolution for discontinuities, and non-oscillatory properties.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
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[1] Abgrall, R., On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, Journal of computational physics, 114, 45-58, (1994) · Zbl 0822.65062
[2] Benson, D.J., Computational methods in Lagrangian and Eulerian hydrocodes, Computer methods in applied mechanics and engineering, 99, 235-394, (1992) · Zbl 0763.73052
[3] Benson, D.J., Momentum advection on a staggered mesh, Journal of computational physics, 100, 143-162, (1992) · Zbl 0758.76038
[4] Campbell, J.C.; Shashkov, M.J., A tensor artificial viscosity using a mimetic finite difference algorithm, Journal of computational physics, 172, 739-765, (2001) · Zbl 1002.76082
[5] Caramana, E.J.; Burton, D.E.; Shashkov, M.J.; Whalen, P.P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, Journal of computational physics, 146, 227-262, (1998) · Zbl 0931.76080
[6] J. Cheng and C.-W. Shu, A high order accurate conservative remapping method on staggered meshes, Applied Numerical Mathematics, in press, doi:10.1016/j.apnum.2007.04.015. · Zbl 1225.76219
[7] Després, B.; Mazeran, C., Lagrangian gas dynamics in two-dimensions and Lagrangian systems, Archive for rational mechanics and analysis, 178, 327-372, (2005) · Zbl 1096.76046
[8] Dukowicz, J.K., A general non-iterative Riemann solver for Godunov method, Journal of computational physics, 61, 119-137, (1985) · Zbl 0629.76074
[9] Dukowicz, J.K.; Cline, M.C.; Addessio, F.L., A general topology Godunov method, Journal of computational physics, 82, 29-63, (1989) · Zbl 0665.76032
[10] Dukowicz, J.K.; Meltz, B.J.A., Vorticity errors in multi-dimensional Lagrangian codes, Journal of computational physics, 99, 115-134, (1992) · Zbl 0743.76058
[11] Erlebacher, G.; Hussaini, M.Y.; Shu, C.-W., Interaction of a shock with a longitudinal vortex, Journal of fluid mechanics, 337, 129-153, (1997) · Zbl 0889.76033
[12] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S.R., Uniformly high order accurate essentially non-oscillatory schemes, III, Journal of computational physics, 71, 231-303, (1987) · Zbl 0652.65067
[13] Harten, A.; Osher, S., Uniformly high-order accurate non-oscillatory schemes I, SIAM journal on numerical analysis, 24, 279-309, (1987) · Zbl 0627.65102
[14] Hirt, C.; Amsden, A.; Cook, J., An arbitrary lagrangian – eulerian computing method for all flow speeds, Journal of computational physics, 14, 227-253, (1974) · Zbl 0292.76018
[15] Jiang, G.; Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 126, 202-228, (1996) · Zbl 0877.65065
[16] Kershaw, D.S.; Prasad, M.K.; Shaw, M.J.; Milovich, J.L., 3D unstructured mesh ALE hydrodynamics with the upwind discontinuous finite element method, Computer methods in applied mechanics and engineering, 158, 81-116, (1998) · Zbl 0954.76045
[17] Koobus, B.; Farhat, C., Second-order time-accurate and geometrically conservative implicit schemes for flow computations on unstructured dynamic meshes, Computer methods in applied mechanics and engineering, 170, 103-129, (1999) · Zbl 0943.76055
[18] Loubère, R.; Shashkov, M.J., A subcell remapping method on staggered polygonal grids for arbitrary-lagrangian – eulerian methods, Journal of computational physics, 209, 105-138, (2005) · Zbl 1329.76236
[19] Luo, H.; Baum, J.D.; Löhner, R., On the computation of multi-material flows using ALE formulation, Journal of computational physics, 194, 304-328, (2004) · Zbl 1136.76401
[20] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM journal on scientific computing, 29, 1781-1824, (2007) · Zbl 1251.76028
[21] Margolin, L.G., Introduction to an arbitrary lagrangian – eulerian computing method for all flow speeds, Journal of computational physics, 135, 198-202, (1997) · Zbl 0938.76067
[22] Munz, C.D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM journal on numerical analysis, 31, 17-42, (1994) · Zbl 0796.76057
[23] von Neumann, J.; Richtmyer, R.D., A method for the calculation of hydrodynamics shocks, Journal of applied physics, 21, 232-237, (1950) · Zbl 0037.12002
[24] Noh, W.F., Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, Journal of computational physics, 72, 78-120, (1987) · Zbl 0619.76091
[25] Peery, J.S.; Carroll, D.E., Multi-material ALE methods in unstructured grids, Computer methods in applied mechanics and engineering, 187, 591-619, (2000) · Zbl 0980.74068
[26] Rogerson, A.; Meiberg, E., A numerical study of the convergence properties of ENO schemes, Journal of scientific computing, 5, 151-167, (1990) · Zbl 0732.65086
[27] Sedov, L.I., Similarity and dimensional methods in mechanics, (1959), Academic Press New York · Zbl 0121.18504
[28] Shu, C.-W., Numerical experiments on the accuracy of ENO and modified ENO schemes, Journal of scientific computing, 5, 127-149, (1990) · Zbl 0732.65085
[29] Shu, C.-W., Preface to the republication of “uniformly high order essentially non-oscillatory schemes, III”, by harten, engquist, osher, and chakravarthy, Journal of computational physics, 131, 1-2, (1997)
[30] C.-W. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in: B. Cockburn, C. Johnson, C.-W. Shu, E. Tadmor (Eds.), Advanced Numerical Approximation of Non-linear Hyperbolic Equations, in: A. Quarteroni (Ed.), Lecture Notes in Mathematics, vol. 1697, Springer, Berlin, 1998, pp. 325-432. · Zbl 0927.65111
[31] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, Journal of computational physics, 77, 439-471, (1988) · Zbl 0653.65072
[32] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, Journal of computational physics, 83, 32-78, (1989) · Zbl 0674.65061
[33] Shu, C.-W.; Zang, T.A.; Erlebacher, G.; Whitaker, D.; Osher, S., High-order ENO schemes applied to two- and three-dimensional compressible flow, Applied numerical mathematics, 9, 45-71, (1992) · Zbl 0741.76052
[34] Smith, R.W., AUSM(ALE): a geometrically conservative arbitrary lagrangian – eulerian flux splitting scheme, Journal of computational physics, 150, 268-286, (1999) · Zbl 0936.76046
[35] Tang, H.Z.; Liu, T.T., A note on the conservative schemes for the Euler equations, Journal of computational physics, 218, 451-459, (2006) · Zbl 1103.76041
[36] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer-Verlag Berlin · Zbl 0923.76004
[37] Woodward, P.R.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, Journal of computational physics, 54, 115-173, (1984) · Zbl 0573.76057
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