# zbMATH — the first resource for mathematics

The construction of compatible hydrodynamics algorithms utilizing conservation of total energy. (English) Zbl 0931.76080
The purpose of this work is to show how the hydrodynamic equations can be differenced compatibly so that they obey the conservation properties. In particular, it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in Lagrangian form, and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system. For staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy. This allows for the specification of forces that can be of an arbitrary complexity, such as those derived from an artificial viscosity or subzonal pressures. These forces originate only in discrete form; nonetheless, the change in internal energy caused by them is still completely determined. The procedure given here is compared to the “method of support operators”, to which it is closely related. Difficulties with conservation of momentum, volume, and entropy are also discussed, together with the proper treatment of boundary conditions and differencing with respect to time. $$\copyright$$ Academic Press.

##### MSC:
 76M99 Basic methods in fluid mechanics 76M20 Finite difference methods applied to problems in fluid mechanics
Full Text:
##### References:
 [1] Burton, D.E., Exact conservation of energy and momentum in staggered-grid hydrodynamics with arbitrary connectivity, Advances in the free Lagrange method, (1990) [2] Samarskii, A.A.; Tishkin, V.F.; Favorskii, A.P.; Shashkov, M.J., Operational finite difference schemes, Differential equations, 17, 854, (1981) · Zbl 0485.65060 [3] Shashkov, M.J., Conservative finite-difference methods on general grids, (1996) · Zbl 0844.65067 [4] 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 [5] E. J. Caramana, M. J. Shashkov, P. P. Whalen, Formulations of artificial viscosity for multi-dimensional shock wave computations, J. Comput. Phys. · Zbl 1392.76041 [6] Landau, L.D.; Lifshitz, E.M., Course in theoretical physics. vol. 1. mechanics, (1960) · Zbl 0080.19702 [7] Benson, D.J., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. methods appl. mech. eng., 99, 235, (1992) · Zbl 0763.73052 [8] Favorskii, A.P., Variational-discrete models of hydrodynamics equations, Differential equations, 16, 834, (1980) [9] Holm, D.D.; Kupershmidt, B.A.; Levermore, C.D., Hamiltonian differencing of fluid dynamics, Adv. in appl. math., 1, 52, (1985) · Zbl 0596.76004 [10] Wilkins, M.L., Calculations of elastic-plastic flow, Methods comput. phys., 3, 211, (1964) [11] Schulz, W.D., Two-dimensional Lagrangian hydrodynamic difference equations, Methods comput. phys., 3, 1, (1964) [12] V. F. Tishkin, N. N. Tiurina, A. P. Favorskii, Finite-difference schemes for calculating hydrodynamic flows in cylindrical coordinates, preprint, No. 23, Keldysh Institute of Applied Mathematics, Moscow, Russia, 1978 [13] Caramana, E.J.; Whalen, P.P., Numerical preservation of symmetry properties of continuum problems, J. comput. phys., 141, 174, (1998) · Zbl 0933.76066 [14] Solov’ev, A.; Shashkov, M., Difference scheme for the Dirichlet particle method in cylindrical method in cylindrical coordinates, conserving symmetry of gas-dynamical flow, Differential equations, 24, 817, (1988) · Zbl 0674.76053 [15] Whalen, P.P., Algebraic limitations on two-dimensional hydrodynamics simulations, J. comput. phys., 124, 46, (1996) · Zbl 0849.76079 [16] Shashkov, M.; Steinberg, S., Solving diffusion equations with rough coefficients in rough grids, J. comput. phys., 129, 383, (1996) · Zbl 0874.65062 [17] Solov’ev, A.; Solov’eva, E.; Tishkin, V.; Favorskii, A.; Shashkov, M., Approximation of finite-difference operators on a mesh of Dirichlet cells, Differential equations, 22, 863, (1986) · Zbl 0616.65021 [18] Mikhailova, N.; Tishkin, V.; Turina, N.; Favorskii, A.; Shashkov, M., Numerical modelling of two-dimensional gas dynamic flows on a variable-structure mesh, U.S.S.R. comput. math. math. phys., 26, 74, (1986) · Zbl 0636.76067 [19] Shashkov, M.; Solov’ov, A., Numerical simulation of two-dimensional flows by the free-Lagrangian method, (1991) [20] Monaghan, J.J., Particle methods for hydrodynamics, Comput. phys. rep., 3, 71, (1985) [21] Dukowicz, J.K.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J. comput. phys., 99, 115, (1992) · Zbl 0743.76058 [22] M. Hyman [23] Sedov, L.I., Similarity and dimensional methods in mechanics, (1959) · Zbl 0121.18504 [24] Burton, D.E., Multidimensional discretizations of conservation laws for unstructured ployhedral grids, (1994) [25] Margolin, L.G.; Adams, T.F., Spatial differencing for finite difference codes, (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.