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A tensor artificial viscosity using a mimetic finite differential algorithm. (English) Zbl 1002.76082
Summary: We develop a two-dimensional tensor artificial viscosity for finite difference shock wave computations. The discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term. The scalar term is based on the form of viscosity first presented by V. F. Kuropatenko [Proc. Steklov Inst. Math. 74 (1966), 116-149 (1967); translation from Tr. Mat. Inst. Steklov. 74, 107-137 (1966; Zbl 0168.46202)], and also contains a limiter designed to switch off the viscosity for shockless compression and rigid-body rotation. Mimetic discretizations are used to derive momentum and energy equations on nonorthogonal grid where the viscosity tensor is evaluated at zone edges. The advantage of tensor viscosity is a reduction of dependence of solution on the relation of grid to flow structure.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
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[1] Benson, D.J., A new two-dimensional flux-limited shock viscosity for impact calculations, Comput. methods appl. mech. eng, 93, 39, (1991) · Zbl 0850.73050
[2] Benson, D.J.; Schoenfeld, S., A total variation diminishing shock viscosity, Comput. mech, 11, 107, (1993) · Zbl 0825.76412
[3] J. C. Campbell, and, M. J. Shashkov, A compatible Lagrangian hydrodynamics algorithm for unstructured grids, LA-UR-00-3231, submitted for publication, available at, http://math.lanl.gov/shashkov/
[4] J. C. Campbell, J. M. Hyman, and, M. J. Shashkov, Mimetic finite difference operators for second-order tensors on unstructured grids, Comput. Math. Appl. in press. · Zbl 0999.65013
[5] Caramana, E.J.; Burton, D.E.; Shashkov, M.J.; Whalen, P.P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. comput. phys, 146, 227, (1998) · Zbl 0931.76080
[6] Caramana, E.J.; Shashkov, M.J., Elimination of artificial grid distortion and hour-Glass type motions by means of Lagrangian subzonal masses and pressures, J. comput. phys, 142, 521, (1998) · Zbl 0932.76068
[7] Caramana, E.J.; Shashkov, M.J.; Whalen, P.P., Formulations of artificial viscosity for multi-dimensional shock wave computations, J. comput. phys, 144, 70, (1998) · Zbl 1392.76041
[8] Fedkiw, R.P.; Marquina, A.; Merriman, B., An isobaric fix for the overheating problem in multimaterial compressible flows, J. comput. phys, 148, 545, (1999) · Zbl 0933.76075
[9] Hyman, J.M.; Shashkov, M., Adjoint operators for the natural discretizations of the divergence, gradient and curl on logically rectangular grids, Appl. numer. math, 25, 413, (1997) · Zbl 1005.65024
[10] Hyman, J.M.; Shashkov, M., Natural discretizations for the divergence, gradient and curl on logically rectangular grids, Comput. math. appl, 33, 81, (1997) · Zbl 0868.65006
[11] V. F. Kurapatenko, in, Difference Methods for Solutions of Problems of Mathematical Physics, I, edited by, N. N. Janenko, Am. Math. Soc. Providence, 1967, p, 116.
[12] R. Landshoff, A Numerical Method for Treating Fluid Flow in the Presence of Shocks, Technical Report LA-1930, Los Alamos National Laboratory, 1955.
[13] L. G. Margolin, A Centered Artificial Viscosity for Cells with Large Aspect Ratios, Technical Report UCRL-53882, Lawrence Livermore National Laboratory, 1988.
[14] Noh, W.F., Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. comput. phys, 72, 78, (1987) · Zbl 0619.76091
[15] Rider, W.J., Revisiting wall heating, J. comput. phys, 162, 395, (2000) · Zbl 0977.76041
[16] Sedov, L.I., Similarity and dimensional methods in mechanics, (1959), Academic Press New York · Zbl 0121.18504
[17] Shashkov, M.; Steinberg, S., Solving diffusion equations with rough coefficients in rough grids, J. comput. phys, 129, 383, (1996) · Zbl 0874.65062
[18] Von Neumann, J.; Richtmyer, R.D., A method for the calculation of hydrodynamic shocks, J. appl. phys, 21, 232, (1950) · Zbl 0037.12002
[19] Wilkins, M.L., Use of artificial viscosity in multidimensional fluid dynamic calculations, J. comput. phys, 36, 281, (1980) · Zbl 0436.76040
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