Hydrocode subcycling stability

Author:
D. L. Hicks

Journal:
Math. Comp. **37** (1981), 69-78

MSC:
Primary 65M10; Secondary 76-04, 76-08

DOI:
https://doi.org/10.1090/S0025-5718-1981-0616360-9

MathSciNet review:
616360

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Abstract: The method of artificial viscosity was originally designed by von Neumann and Richtmyer for calculating the propagation of waves in materials that were hydrodynamic and rate-independent (e.g., ideal gas law). However, hydrocodes (such as WONDY) based on this method continue to expand their repertoire of material laws even unto material laws that are rate-dependent (e.g., Maxwell's material law). Restrictions on the timestep required for stability with material laws that are rate-dependent can be considerably more severe than restrictions of the Courant-Friedrichs-Lewy (CFL) type that are imposed in these hydrocodes. These very small timesteps can make computations very expensive. An alternative is to go ahead and integrate the conservation laws with the usual CFL timestep while subcycling (integrating with a smaller timestep) the integration of the stress-rate equation. If the subcycling is done with a large enough number of subcycles (i.e., with a small enough subcycle timestep), then the calculation is stable. Specifically, the number of subcycles must be one greater than the ratio of the CFL timestep to the relaxation time of the material.

**[1]**J. Asay, D. Hicks & D. Holdridge, "Comparison of experimental and calculated elastic-plastic wave profiles in LiF,"*J. Appl. Phys.*, v. 46, 1975, pp. 4316-4322.**[2]**D. L. Hicks,*Stability analysis of WONDY (a hydrocode based on the artificial viscosity method of von Neumann and Richtmyer) for a special case of Maxwell’s law*, Math. Comp.**32**(1978), no. 144, 1123–1130. MR**0483944**, https://doi.org/10.1090/S0025-5718-1978-0483944-0**[3]**W. Herrmann, P. Holzhauser & R. Thompson,*WONDY*:*A Computer Program for Calculating Problems of Motion in One Dimension*, Sandia Laboratories Report SC-RR-66-601, 1966.**[4]**Robert D. Richtmyer and K. W. Morton,*Difference methods for initial-value problems*, Second edition. Interscience Tracts in Pure and Applied Mathematics, No. 4, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1967. MR**0220455****[5]**J. Von Neumann and R. D. Richtmyer,*A method for the numerical calculation of hydrodynamic shocks*, J. Appl. Phys.**21**(1950), 232–237. MR**0037613**

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0616360-9

Article copyright:
© Copyright 1981
American Mathematical Society