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Conservative smoothing stabilizes discrete-numerical instabilities in SPH material dynamics computations. (English) Zbl 0882.76064

Summary: We give an analysis of smoothed particle hydrodynamics (SPH) which reveals that SPH: (i) has an instability; (ii) cannot be stabilized with artificial viscosities; (iii) can be stabilized with conservative smoothing.

MSC:

76M25 Other numerical methods (fluid mechanics) (MSC2010)
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[1] Campbell, P. M., Some new algorithms for boundary value problems in smooth particle hydrodynamics, Mission Research Corp. Tech. Rep., DNA-TR-88-286 (1988)
[2] Courant, R.; Friedrichs, K.; Lewy, H., Über die partiellen Differenzengleichungen der mathematischen Physik, Math. Ann., 100, 32-74 (1928) · JFM 54.0486.01
[3] Guenther, C.; Hicks, D. L.; Swegle, J. W., Conservative Smoothing versus artificial viscosity, SNLA SAND94-1853 (1994)
[4] Gel’fand, I. M.; Shilov, G. E., (Generalized Functions, Vol. 1 (1964), Academic Press) · Zbl 0115.33101
[5] Gingold, R. A.; Monaghan, J. J., Smoothed particle hydrodynamics: theory and applications to nonspherical stars, Mon. Not. Astr. Soc., 181, 375-389 (1977) · Zbl 0421.76032
[6] Hadamard, J. S., Lectures on Cauchy’s Problem in Linear Partial Differential Equations (1923), Yale U. Press · JFM 49.0725.04
[7] F. Harlow, The Particle-In-Cell (PIC) Method for Numerical Solutions of Problems of Fluid Mechanics, Proc. Symp. in Appl. Math.; F. Harlow, The Particle-In-Cell (PIC) Method for Numerical Solutions of Problems of Fluid Mechanics, Proc. Symp. in Appl. Math.
[8] Herrmann, W., A Lagrangean finite difference method for two-dimensional motion including material strength, AFWL-TR-64-107 (1964)
[9] Hicks, D. L., The Convergence of Numerical Solutions of Hydrodynamic Shock Problems, AFWL-TR-69-20 (1969), CHFS AD849487
[10] Hicks, D. L., Stability Analysis of WONDY (a Hydrocode based on the Artificial Viscosity Method of von Neumann and Richtmyer) for a Special Case of Maxwell’s Material Law, Math. Comp., 32, 144, 1123-1130 (1978) · Zbl 0388.76006
[11] Hicks, D. L.; Kuttler, K. L., Error bounds for numerical solutions of hydrodynamical problems including shocks, SIAM J. Num. Analy., 28, 3, 662-684 (1991) · Zbl 0733.76035
[12] Hicks, D. L.; Liebrock, L. M.; Wen, Y., Eigenform error estimates for computational material dynamics with shocks, Part 1.1 Hooke’s law materials, Appl. Math. & Comp., 66, 2 & 3, 181-216 (1994) · Zbl 0818.73074
[13] Hicks, D. L.; Swegle, J. W.; Attaway, S. W., SPH: Instabilities, Wall Heating, and Conservative Smoothing, (Wingate, C.; Miller, W., Proc. Workshop on Advances in SPH. Proc. Workshop on Advances in SPH, Los Alamos National Lab., Los Alamos, NM, Los Alamos Rep. LA-UR-93-4375 (21-23 Sept. 1993)), 223-256 · Zbl 0879.76073
[14] Kolsky, H., A method for the numerical solution of transient hydrodynamic shock problems in two space dimensions, Los Alamos Rep. LA-1867 (1955)
[15] Landshoff, R., A numerical method for treating fluid flow in the presence of shocks, Los Alamos. Rep., LA-1930 (1955)
[16] Lax, P. D.; Richtmyer, R. D., Survey of the stability of linear finite difference equation, Comm. Pure Appl. Math., 9, 267 (1956) · Zbl 0072.08903
[17] Lucy, L. B., A numerical approach to the testing of the fission hypothesis, Astr. J., 82, 1013-1024 (1977)
[18] Monaghan, J.; Gingold, R., Shock simulation by the particle method SPH, J. Comp. Physics., 52, 374-389 (1983) · Zbl 0572.76059
[19] von Neumann, J.; Richtmyer, R., A method for the numerical calculation of hydrodynamic shocks, J. Appl. Physics, 21, 232-237 (1950) · Zbl 0037.12002
[20] Ransom, V. H.; Hicks, D. L., Hyperbolic two-pressure models for two-phase flow, J. Comp. Physics, 53, 1, 124-151 (1984) · Zbl 0537.76070
[21] Richtmyer, R.; Morton, K., Difference Methods for Initial Value Problems (1967), Interscience Publishers · Zbl 0155.47502
[22] Swegle, J. W.; Attaway, S. W.; Heinstein, M. W.; Mello, F. J.; Hicks, D. L., An Analysis of Smoothed Particle Hydrodynamics, SNLA SAND93-2513 (1993)
[23] Swegle, J. W.; Hicks, D. L., Stability and consistency of the SPH Method, (Wingate, C.; Miller, W., Proc. Workshop on Advances in SPH. Proc. Workshop on Advances in SPH, Los Alamos National Lab., Los Alamos, NM, Los Alamos Rep. LA-UR-93-4375 (21-23 Sept. 1993)), 189-222
[24] Swegle, J. W.; Hicks, D. L.; Attaway, S. W., Stability Analysis of SPH, J. Comp. Physics, 116, 123-134 (1995) · Zbl 0818.76071
[25] Wen, Y.; Hicks, D. L.; Swegle, J. W., Stabilizing SPH with Conservative Smoothing, SNLA Report SAND94-1932 (1994)
[26] Whitaker, W.; Nawrocki, E.; Needham, C.; Troutman, W., Theoretical Calculations of the phenomenology of HE detonations, AFWL-TR-66-141 (1966)
[27] Wilkins, M., Calculation of elastic-plastic flow, Lawrence Livermore Rep., UCRL-7322 (1963)
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