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Computation of steady shocks by second-order finite-difference schemes

Author: Lasse K. Karlsen
Journal: Math. Comp. 34 (1980), 391-400
MSC: Primary 65M10; Secondary 76L05
MathSciNet review: 559192
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Abstract: The computational stability of steady shocks which satisfy the entropy condition is considered for the scalar conservation law \[ \frac {{\partial u}}{{\partial t}} + \frac {\partial }{{\partial x}}\left ( {\frac {1}{2}{u^2}} \right ) = 0.\] It is shown that the computation of the pure initial value problem by Lax-Wendroff type schemes approaches a steady state if the initial data satisfies a specified condition, and that this condition is always satisfied for the corresponding initial-boundary value problem with a finite number of grid points. The effect of machine accuracy on the influence of the boundaries on the error near the shock is also discussed.

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  • A. Harten, J. M. Hyman, and P. D. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), no. 3, 297–322. With an appendix by B. Keyfitz. MR 413526, DOI
  • O. A. Oleĭnik, Discontinuous solutions of non-linear differential equations, Amer. Math. Soc. Transl. (2) 26 (1963), 95–172. MR 0151737, DOI
  • Heinz-Otto Kreiss, Difference approximations for the initial-boundary value problem for hyperbolic differential equations, Numerical Solutions of Nonlinear Differential Equations (Proc. Adv. Sympos., Madison, Wis., 1966) John Wiley & Sons, Inc., New York, 1966, pp. 141–166. MR 0214305
  • R. D. RICHTMEYER & K. W. MORTON, Difference Methods for Initial Value Problems, Interscience, New York, 1967. R. W. MacCORMACK, The Effect of Viscosity in Hypervelocity Impact Cratering, AIAA Paper 69-354, 1969. L. K. KARLSEN, A Criterion for the Existence of Erroneous Modes Near Steady Shocks in Conservative Finite-Difference Computations, Trans. Roy. Inst. Techn., TRITA-FPT-009, Stockholm, 1974.
  • R. F. Warming and Richard M. Beam, Upwind second-order difference schemes and applications in aerodynamic flows, AIAA J. 14 (1976), no. 9, 1241–1249. MR 459301, DOI

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Article copyright: © Copyright 1980 American Mathematical Society