Three-dimensional second-order accurate difference schemes for discontinuous hydrodynamic flows

Authors:
Ephraim L. Rubin and Stanley Preiser

Journal:
Math. Comp. **24** (1970), 57-63

MSC:
Primary 76.65

MathSciNet review:
0264904

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Abstract: In this paper, we show how a second-order accurate two-step method used in the numerical computation of hydrodynamic flows may be derived directly from the integral conservation laws. A necessary and sufficient condition for stability for the linearized equations is derived for the three-dimensional Cartesian coordinate case.

**[1]**J. L. Anderson, S. Preiser & E. L. Rubin, "Conservation form of the equations of hydrodynamics in curvilinear coordinate systems,"*J. Computational Phys.*, v. 2, 1968, pp. 279-287.**[2]**Peter Lax and Burton Wendroff,*Systems of conservation laws*, Comm. Pure Appl. Math.**13**(1960), 217–237. MR**0120774****[3]**R. D. Richtmyer,*A Survey of Difference Methods for Non-Steady Fluid Dynamics*, NCAR Technical Note 63-2, 1962.**[4]**E. L. Rubin & S. Z. Burstein, "Difference methods for the inviscid and viscous equations of a compressible gas,"*J. Computational Phys.*, v. 2, 1967, pp. 178-196.**[5]**C. Truesdell and R. Toupin,*The classical field theories*, Handbuch der Physik, Bd. III/1, Springer, Berlin, 1960, pp. 226–793; appendix, pp. 794–858. With an appendix on tensor fields by J. L. Ericksen. MR**0118005****[6]**A. H. Taub,*On the Derivation of Difference Equations for Hydrodynamics*, Los Alamos Scientific Laboratory Report LA-2073, 1957.**[7]**Luther Pfahler Eisenhart,*Continuous groups of transformations*, Dover Publications, Inc., New York, 1961. MR**0124008****[8]**Heinz-Otto Kreiss,*On difference approximations of the dissipative type for hyperbolic differential equations*, Comm. Pure Appl. Math.**17**(1964), 335–353. MR**0166937**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0264904-4

Keywords:
Fluid dynamics,
stability,
conservation laws

Article copyright:
© Copyright 1970
American Mathematical Society