## Studies in numerical nonlinear instability. II. A new look at $$u_ t+uu_ x=0$$.(English)Zbl 0612.65053

[For part I by the second author see SIAM J. Sci. Stat. Comput. 6, 923- 938 (1985; reviewed above).]
It is shown for a leap-frog discretization of equation $$u_ t+uu_ x$$ that any particular unstable solution behaves as an attractor of other solutions. The overflow time is estimated and related to the notions of stability threshold and restricted stability.
Reviewer: V.A.Kostova

### MSC:

 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 35L60 First-order nonlinear hyperbolic equations

Zbl 0612.65052
Full Text:

### References:

 [1] Aref, H.; Daripa, P.K., SIAM J. sci. stat. comput., 5, 856, (1984) [2] Briggs, W.L.; Newell, A.C.; Sarie, T., J. comput. phys., 51, 83, (1983) [3] Fornberg, B., Math. comput., 27, 45, (1973) [4] {\scJ. C. López-Marcos and J. M. Sanz-Serna}, IMA J. Numer. Anal., in press. [5] Majda, A.; Osher, S., Numer. math., 30, 429, (1978) [6] Palencia, C.; Sanz-Serna, J.M., IMA J. numer. anal., 4, 109, (1984) [7] Palencia, C.; Sanz-Serna, J.M., Numer. math., 44, 279, (1984) [8] Richtmyer, R.; Morton, K.W., Difference methods for initial value problems, (1967), WileyInterscience New York · Zbl 0155.47502 [9] Sanz-Serna, J.M., SIAM J. sci. slat. comput., 6, 923, (1985) [10] Sanz-Serna, J.M., (), 64 [11] Sanz-Serna, J.M.; Palencia, C., Math. comput., 45, 143, (1985) [12] {\scJ.M. Sanz-Serna and F. Vadillo}Proceedings Dundee 1985, edited by D. F. Griffiths and G.A. Watson (Pitman, London, in press). [13] {\scJ. M. Sanz-Serna and F. Vadillo}, Studies in numerical nonlinear instability III: Augmented Hamiltonian systems SIAM J. Appl. Math., in press. · Zbl 0632.65129 [14] Sanz-Serna, J.M.; Verwer, J., J. math. anal. appl., 116, 456, (1986) [15] Vadillo, F., Thesis, (1985), (unpublished)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.