Abstract
Let the two dimensional scalar advection equation be given by
We prove that the stability region of the MacCormack scheme for this equation isexactly given by
where Δ t , Δ x and Δ y are the grid distances. It is interesting to note that the stability region is identical to the one for Lax-Wendroff scheme proved by Turkel.
Zusammenfassung
Wir betrachten die zweidimensionale skalare Advektionsgleichung
und zeigen, daß der Stabilitätsbereich des MacCormack-Schemasgenau durch
gegeben ist, wo Δ t , Δ x and δ y die Gitterabstände sind. Interessanterweise ist dieser Stabilitätsbereich identisch mit dem von Turkel für das Lax-Wendroff-Schema bestimmten Stabilitätsbereich.
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References
Collins, G. E.: Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In: Lecture Notes in Computer Science, pp. 134–183. Berlin, Heidelberg, New York: Springer 1975.
Collins, G. E., Hong, H.: Partial cylindrical algrebaic decomposition for quantifier elimination. J. Symb. Comput.12, 229–328 (1991).
Hong, H.: An improvement of the projection operator in cylindrical algebraic decomposition. In: International Symposium of Symbolic and Algebraic Computation ISSAC-90, pp. 261–264, 1990.
Hong, H.: Improvements in CAD-based quantifier elimination. PhD thesis, The Ohio State University, 1990.
Hong, H.: Simple solution formula construction in cylindrical algebraic decomposition based quantifier elimination. In: International Conference on Symbolic and Algebraic Computation ISSAC-92, pp. 177–188, 1992.
Lax, P. D., Wendroff, B.: Difference schemes for hyperbolic equations with higher order of accuracy. Comm. Pure Appl. Math17, 381–398 (1964).
Liska, R., Steinberg, S.: Applying quantifier elimination to stability analysis of difference schemes. Comput. J.36, 497–503 (1993).
MacCormack, R.: The effect of viscosity in hypervelocity impact cratering. AIAA Paper No. 69-354 (1969).
Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. J. Symb. Comput.13, 255–352 (1992).
Tarski, A.: A decision method for elementary algebra and geometry, 2nd ed. Berkeley: University of California Press, 1951.
Turkel, E.: Symmetric hyperbolic difference schemes and matrix problems. Lin Alg. Appl.16, 109–129 (1977).
Wendroff, B.: The stability of MacCormack's method for the scalar advection equations. Appl. Math. Lett.4, 89–91 (1991).
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This research was done within the framework of the European project ACCLAIM.
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Hong, H. The exact region of stability for MacCormack scheme. Computing 56, 371–383 (1996). https://doi.org/10.1007/BF02253461
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DOI: https://doi.org/10.1007/BF02253461