# zbMATH — the first resource for mathematics

Formulations of artificial viscosity for multi-dimensional shock wave computations. (English) Zbl 1392.76041
Summary: In this paper we present a new formulation of the artificial viscosity concept. Physical arguments for the origins of this term are given and a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated. The first important property is that by definition a viscosity must always be dissipative, transferring kinetic energy into internal energy, and must never act as a false pressure. The artificial viscous force should be Galilean invariant and vary continuously as a function of the criterion used to determine compression and expansion, and remain zero for the latter case. These requirements significantly constrain the functional form that the artificial viscous force can have. In addition, an artificial viscosity should be able to distinguish between shock-wave and adiabatic compression, and not result in spurious entropy production when only the latter is present. It must therefore turn off completely for self-similiar motion, where only a uniform stretching and/or a rigid rotation occurs. An additional important, but more subtle, condition where the artificial viscosity should produce no effect is along the direction tangential to a convergent shock front, since the velocity is only discontinuous in the normal direction. Our principal result is the development of a new formulation of an edge-centered artificial viscosity that is to be used in conjunction with a staggered spatial placement of variables that meets all of these standards, and without the need for problem dependent numerical coefficients that have in the past made the artificial viscosity method appear somewhat arbitrary. Our formulation and numerical results are given with respect to two spatial dimensions but all of our arguments carry over directly to three dimensions. A central feature of our development is the implementation of simple advection limiters in a straightforward manner in more than one dimension to turn off the artificial viscosity for the above mentioned conditions, and to substantially reduce its effect when strong velocity gradients are absent.

##### MSC:
 76M20 Finite difference methods applied to problems in fluid mechanics 76L05 Shock waves and blast waves in fluid mechanics 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
Full Text:
##### References:
 [1] VonNeumann, J.; Richtmyer, R. D., A method for the numerical calculation of hydrodynamic shocks, J. Appl. Phys., 21, 232, (1950) · Zbl 0037.12002 [2] E. J. Caramana, D. E. Burton, M. J. Shashkov, P. P. Whalen, The development of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys. · Zbl 0931.76080 [3] Schulz, W. D., Two-dimensional Lagrangian hydrodynamic difference schemes, Methods Comput. Phys., 3, 1, (1964) [4] Christiansen, R. B., Godunov Methods on a Staggered Mesh—An Improved Artificial Viscosity, (1991) [5] Benson, D. J.; Schoenfeld, S., A total variation diminishing shock viscosity, Comput. Mech., 11, 107, (1993) · Zbl 0825.76412 [6] Landshoff, R., A Numerical Method for Treating Fluid Flow in the Presence of Shocks, (1955) [7] Wilkins, M. L., Use of artificial viscosity in multidimensional shock wave problems, J. Comput. Phys., 36, 281, (1980) · Zbl 0436.76040 [8] A. P. Favorskii, L. V. Moiseenko, V. F. Tishkin, N. N. Tyurina, The introduction of artificial dissipators into finite-difference schemes of hydrodynamics, M. V. Keldysh Institute of Applied Mathematics, Moscow, Russia, 1982 [9] Mikhailova, N. V.; Tishkin, V. F.; Turina, N. N.; Favorskii, A. P.; Shashkov, M. Yu., Numerical modelling of two-dimensional gas-dynamic flows on a variable structure mesh, USSR Comput. Math. and Math. Phys., 26, 74, (1986) · Zbl 0636.76067 [10] Richtmyer, R. D.; Morton, K. W., Difference Methods for Initial-Value Problems, 313, (1967) · Zbl 0155.47502 [11] van Leer, B., Towards the ultimate conservative difference scheme 5: A second order sequel to Godunov’s method, J. Comput. Phys., 32, 101, (1979) · Zbl 1364.65223 [12] Dukowicz, J. K., A general, non-iterative reimann solver for Godunov’s method, J. Comput. Phys., 61, 119, (1985) · Zbl 0629.76074 [13] Kuropatenko, V. F., Difference Methods for Solutions of Problems of Mathematical Physics, 1, 116, (1967) [14] Mihalas, D.; Mihalas, B., Foundations of Radiation Hydrodynamics, 283, (1984) · Zbl 0651.76005 [15] Landau, L. D.; Lifshitz, E. M., Course of theoretical physics, Fluid Mechanics, 6, 48, (1982) [16] Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49, 357, (1983) · Zbl 0565.65050 [17] Burton, D. E., Exact conservation of energy and momentum in staggered-grid hydrodynamics with arbitrary connectivity, Advances in the Free Lagrange Method, (1990) [18] W. D. Schulz, 1979 [19] Caramana, E. J.; Shashkov, M. J., Elimination of artificial grid distortion and hourglass-type motions by means of Lagrangian subzonal masses and pressures, J. Comput. Phys., 142, 521, (1998) · Zbl 0932.76068 [20] Burton, D. E., Multidimensional Discretization of Conservation Laws for Unstructured Polyhedral Grids, (1994) [21] D. E. Burton [22] Caramana, E. J.; Whalen, P. P., Numerical preservation of symmetry properties of continuum problems, J. Comput. Phys., 141, 174, (1998) · Zbl 0933.76066 [23] Margolin, L. G., A Centered Artificial Viscosity for Cells with Large Aspect Ratio, (1988) [24] 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 [25] Coggeshall, S. V.; Meyer-ter-Vehn, J., Group invariant solutions and optimal systems for multidimensional hydrodynamics, J. Math. Phys., 33, 3585, (1992) · Zbl 0763.76006 [26] Sedov, L. I., Similarity and Dimensional Methods in Mechanics, (1959) · Zbl 0121.18504 [27] Cameron, I. G., An analysis of the errors caused by using artificial viscosity terms to represent steady-state flow, J. Comput. Phys., 1, 1, (1966) · Zbl 0148.45501 [28] Margolin, L. G.; Ruppel, H. M.; Demuth, R. B., Gradient Scaling for Nonuniform Meshes, (1985)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.