## 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
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### 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)
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