×

An instability of the Godunov scheme. (English) Zbl 1122.35074

Authors’ summary: We construct a solution to a 2 \(\times 2\) strictly hyperbolic system of conservation laws, showing that the Godunov scheme can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing-viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or \(L^{1}\)-stability estimates can in general be valid for finite difference schemes.

MSC:

35L65 Hyperbolic conservation laws
35A35 Theoretical approximation in context of PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Baiti, Discrete Contin Dyn Syst 13 pp 877– (2005)
[2] Bianchini, Arch Ration Mech Anal 167 pp 1– (2003)
[3] Bianchini, Comm Pure Appl Math 59 pp 688– (2006)
[4] Bianchini, Ann of Math (2) 161 pp 223– (2005)
[5] Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem. Oxford Lecture Series in Mathematics and Its Applications, 20. Oxford University Press, Oxford, 2000. · Zbl 0997.35002
[6] Bressan, Chinese Ann Math Ser B 21 pp 269– (2000)
[7] Bressan, Arch Ration Mech Anal 149 pp 1– (1999)
[8] Bressan, Discrete Contin Dynam Systems 6 pp 21– (2000)
[9] Ding, Comm Math Phys 121 pp 63– (1989)
[10] DiPerna, Arch Rational Mech Anal 82 pp 27– (1983)
[11] An introduction to probability theory and its applications. Vol. I. 3rd ed. Wiley, New York-London-Sydney, 1968.
[12] Glimm, Comm Pure Appl Math 18 pp 697– (1965)
[13] Godunov, Mat Sb (NS) 47 (89) pp 271– (1959)
[14] Jennings, Comm Pure Appl Math 27 pp 25– (1974)
[15] Lax, Comm Pure Appl Math 7 pp 159– (1954)
[16] Lax, Comm Pure Appl Math 10 pp 537– (1957)
[17] Numerical methods for conservation laws. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 1990. · Zbl 0723.65067
[18] LeVeque, Trans Amer Math Soc 288 pp 115– (1985)
[19] Liu, Comm Pure Appl Math 52 pp 1553– (1999)
[20] Majda, Comm Pure Appl Math 32 pp 445– (1979)
[21] Serre, Mat Contemp 11 pp 153– (1996)
[22] Yang, Proc Amer Math Soc 131 pp 1257– (2003)
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.