Abstract
Two-dimensional systems of linear hyperbolic equations are studied with regard to their behavior under a solution strategy that in alternate time-steps exactly solves the component one-dimensional operators. The initial data is a step function across an oblique discontinuity. The manner in which this discontinuity breaks up under repeated applications of the split operator is analyzed, and it is shown that the split solution will fail to match the true solution in any case where the two operators do not share all their eigenvectors. The special case of the fluid flow equations is analyzed in more detail, and it is shown that arbitrary initial data gives rise to “pseudo acoustic waves” and a nonphysical stationary wave. The implications of these find-ings for the design of high-resolution computing schemes are discussed.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Strang, “On the Construction and Comparison of Difference Schemes,” SIAMI Numer. Anal, 5, 506 (1968).
H.C. Yee, “Upwind and Symmetric Shock-Capturing Schemes,” Proceedings Seminar on Computational Aerodynamics, M. M. Hafez, Ed., University of California at Davis, AIAA Special Publication, Spring 1986.
P. L. Roe, “Characteristic-Based Schemes for the Euler Equations,” Ann. Rev. Fluid Mech.337 (1986).
P. Colella, “Glimm’s Method for Gas Dynamics,” SIAMI Sci. Statist. Comput., 3, 77 (1982).
M. Crandall and A. Majda, “The Method of Fractional Steps for Conservation Laws,” Numer. Math., 34, 285 (1980).
R.C.Y. Chin and G.W. Hedstrom, “A Dispersion Analysis for Difference Schemes: Tables of Generalized Airy Functions,” Math. Comp., 32, 1163 (1978).
L.J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
A. Harten, “On a Large Time-Step High-Resolution Scheme,” ICASE Report No. 82, 34, 1982.
R. LeVeque, “A Large Time-Step Generalization of Godunov’s Method for Systems of Conservation Laws,” SIAMI Numer. Anal., 22, 1051 (1985).
A. Harten, “The Artificial Compression Method for Computation of Shocks and Contact Discontinuities. III. Self-Adjusting Hybrid Schemes,” Math. Comp, 32, 363 (1978).
P. L. Roe and M. J. Baines, “Asymptotic Behavior of Some Nonlinear Schemes for Linear Advection,” in Notes on Numerical Fluid Mechanics, M. Pandolfi and R. Piva (Eds.), Vieweg, Braunsschweig, 1984. pp. 283 – 290.
H. C. Yee and J. L. Shinn, “Semi-Implicit and Fully Implicit Shock-Capturing Methods for Hyperbolic Conservation Laws with Stiff Source Terms,” AIAA Report No. 87, 1116, July 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Roe, P.L. (1997). Discontinuous Solutions to Hyperbolic Systems Under Operator Splitting. In: Hussaini, M.Y., van Leer, B., Van Rosendale, J. (eds) Upwind and High-Resolution Schemes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60543-7_19
Download citation
DOI: https://doi.org/10.1007/978-3-642-60543-7_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-64452-8
Online ISBN: 978-3-642-60543-7
eBook Packages: Springer Book Archive