NORMAS REFERIDAS AL USO INDEBIDO DEL PODER PÚBLICO (CORRUPCIÓN)

In order to motivate the use of a semi-implicit scheme to solve (3.26-3.28), here we perform a von- Neumann stability analysis of a "natural/reasonable" explicit scheme and show that it leads to in- stabilities. Since such an analysis requires a constant coefficient linear system, we consider here the linearized, unsteady, transonic small disturbance equations

ut+ ux+ vy = 0, (A.1)

vx = uy. (A.2)

We discretize these equations using forward finite differences in t and x, and centered finite differences in y. This leads to the scheme

U_i,jn+1 = U_i,jn − ∆t _Un i,j− Ui−1,jn ∆x + V_i,j+1n − Vn i,j−1 2∆y  , (A.3) V_i,jn = V_i+1n − ∆x _Un i,j+1− Ui,j−1n 2∆y  . (A.4)

Then we compute the (periodic) discrete eigenfunctions for the scheme using the ansatz

U_p,qn = AGnei(kxp+lyq)_{, (A.5)}

where k and l are the discrete wave numbers, G is the growth factor, A and B are constants, x_p = p∆x, yq = q∆y, and

U_p,qn = U (xq, yp, tn), (A.7)

V_p,qn = V (xq, yp, tn). (A.8)

In the standard fashion of the von Neumann stability analysis, this leads to an eigenvalue problem for the vector with components A and B, with eigenvalue G. Solving this problem yields

G = 1 − ∆t ∆x 1 − e−ik∆x ∆x − ∆t∆x ∆y2 sin2(l∆y) (1 − eik∆x₎. (A.9)

Note that, in (A.9), the term

∆t∆x ∆y2

sin2(l∆y)

(1 − eik∆x₎ (A.10)

can be traced back to the explicit treatment of vy. This term causes instability, since it can become

arbitrarily large for k∆x small and sin2(l∆y) away from zero, independently of the size of ∆t. Hence, the scheme is unstable. Instabilities like this one are inevitable in explicit finite difference schemes, regardless of the choice. The reason is that the wave speeds in the y-direction are unbounded. No explicit scheme can hence satisfy the CFL (Courant-Friedrichs-Lewy) condition.

References

[1] B. Barker, J. Humpherys, G. Lyng, and K. Zumbrun. Viscous hyperstabilization of detonation waves in one space dimension. arXiv preprint arXiv:1311.6417, 2013.

[2] J. B. Bdzil and D. S. Stewart. Theory of detonation shock dynamics. In Shock Waves Science and Technology Library, Vol. 6, pages 373–453. Springer, 2012.

[3] A. Bourlioux, A. J. Majda, and V. Roytburd. Theoretical and numerical structure for unstable one-dimensional detonations. SIAM Journal of Applied Mathematics, 51:303–343, 1991.

[4] P. Clavin and F. A. Williams. Dynamics of planar gaseous detonations near Chapman-Jouguet conditions for small heat release. Combustion Theory and Modelling, 6(1):127–139, 2002.

[5] P. Clavin and F. A. Williams. Analytical studies of the dynamics of gaseous detonations. Philo- sophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 370(1960):597–624, 2012.

[6] J. J. Erpenbeck. Stability of idealized one-reaction detonations. Phys. Fluids, 7:684–696, 1964.

[7] L. M. Faria, A. R. Kasimov, and R. R. Rosales. Study of a model equation in detonation theory. SIAM Journal on Applied Mathematics, 74(2):547–570, 2014.

[8] W. Fickett. Detonation in miniature. American Journal of Physics, 47(12):1050–1059, 1979.

[9] W. Fickett and W. C. Davis. Detonation: theory and experiment. Dover Publications, 2011.

[10] H. M. Glaz, P. Colella, I. I. Glass, and R. L. Deschambault. A numerical study of oblique shock- wave reflections with experimental comparisons. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 398(1814):117–140, 1985.

[11] A. K. Henrick, T. D. Aslam, and J. M. Powers. Simulations of pulsating one-dimensional detona- tions with true fifth order accuracy. J. Comput. Phys., 213(1):311–329, 2006.

[12] J. Humpherys, G. Lyng, and K. Zumbrun. Stability of viscous detonations for Majda’s model. Physica D: Nonlinear Phenomena, 259:63–80, 2013.

[13] J. K. Hunter. Asymptotic equations for nonlinear hyperbolic waves. In Surveys in applied math- ematics, pages 167–276. Springer, 1995.

[14] J. K. Hunter and M. Brio. Weak shock reflection. J. Fluid Mech., 410:235–261, 2000.

[15] A. R. Kasimov, L. M. Faria, and R. R. Rosales. Model for shock wave chaos. Physical Review Letters, 110(10):104104, 2013.

[16] A. R. Kasimov and D. S. Stewart. On the dynamics of self-sustained one-dimensional detonations: A numerical study in the shock-attached frame. Physics of Fluids, 16:3566, 2004.

[17] J. B. Keller. Rays, waves and asymptotics. Bulletin of the American mathematical society, 84(5):727–750, 1978.

[18] D. I. Ketcheson, K. T. Mandli, A. J. Ahmadia, A. Alghamdi, M. Quezada de Luna, M. Parsani, M. G. Knepley, and M. Emmett. PyClaw: Accessible, Extensible, Scalable Tools for Wave Prop- agation Problems. SIAM Journal on Scientific Computing, 34(4):C210–C231, November 2012. [19] H. I. Lee and D. S. Stewart. Calculation of linear detonation instability: One-dimensional insta-

bility of plane detonation. J. Fluid Mech., 212:103–132, 1990.

[20] J. H. S. Lee. The Detonation Phenomenon. Cambridge University Press, 2008.

[21] R. J. LeVeque. Finite volume methods for hyperbolic problems. Cambridge University Press, 2002. [22] C. C. Lin, E. Reissner, and H. S. Tsien. On two-dimensional non-steady motion of a slender body

in a compressible fluid. Journal of Mathematics and Physics, 27(3):220, 1948.

[23] A. Majda. A qualitative model for dynamic combustion. SIAM Journal on Applied Mathematics, 41(1):70–93, 1980.

[24] H. Ng, A. Higgins, C. Kiyanda, M. Radulescu, J. Lee, K. Bates, and N. Nikiforakis. Nonlinear dynamics and chaos analysis of one-dimensional pulsating detonations. Combust. Theory Model, 9(1):159–170, 2005.

[25] E. Oran and J. P. Boris. Numerical simulation of reactive flow. Cambridge University Press, Cambridge, UK, 2001.

[26] M. I. Radulescu and J. Tang. Nonlinear dynamics of self-sustained supersonic reaction waves: Fickett’s detonation analogue. Phys. Rev. Lett., 107(16), 2011.

[27] R. R. Rosales. Diffraction effects in weakly nonlinear detonation waves. In Nonlinear Hyperbolic Problems, pages 227–239. Springer, 1989.

[28] R. R. Rosales. An introduction to weakly nonlinear geometrical optics. In Multidimensional Hyperbolic Problems and Computations, pages 281–310. Springer, 1991.

[29] R. R. Rosales and A. J. Majda. Weakly nonlinear detonation waves. SIAM Journal on Applied Mathematics, 43(5):1086–1118, 1983.

[30] M. Short. Multidimensional linear stability of a detonation wave at high activation energy. SIAM Journal on Applied Mathematics, 57(2):307–326, 1997.

[31] M. Short and D. S. Stewart. Cellular detonation stability. Part 1. A normal-mode linear analysis. Journal of Fluid Mechanics, 368:229–262, 1998.

[32] M. Short and D. S. Stewart. The multi-dimensional stability of weak-heat-release detonations. Journal of Fluid Mechanics, 382:109–135, 1999.

[33] D. S. Stewart, T. D. Aslam, and J. Yao. On the evolution of cellular detonation. In Proceedings of the Combustion Institute, volume 26, pages 2981–2989, Pittsburgh, PA, 1996. The Combustion Institute.

[34] S. H. Strogatz. Nonlinear dynamics and chaos: With applications to physics, biology, chemistry, and engineering. Westview Press, 1994.

[35] E. G. Tabak and R. R. Rosales. Focusing of weak shock waves and the von Neumann paradox of oblique shock reflection. Physics of Fluids (1994-present), 6(5):1874–1892, 1994.

[36] B. D. Taylor, A. R. Kasimov, and D. S. Stewart. Mode selection in weakly unstable two- dimensional detonations. Combustion Theory and Modelling, 13(6):973–992, 2009.

[37] B. V. Voitsekhovskii, V. V. Mitrofanov, and M. Y. Topchian. The Structure of Detonation Front in Gases. Report FTD-MTD-64-527. Foreign Technology Division, Wright Patterson Air Force Base, OH (AD 633-821)., 1966.

[38] J. von Neumann. Oblique reflection of shocks. In Collected Works, volume VI, pages 238–299. Pergamon, New York, 1963.

[39] G. B. Whitham. Linear and Nonlinear Waves. John Wiley and Sons, New York, NY, 1974. [40] F. A. Williams. Combustion theory. Westview Press, 1985.

[41] J. Yao and D. S. Stewart. On the dynamics of multi-dimensional detonation. J. Fluid Mech., 309:225–275, 1996.

[42] E. A. Zabolotskaya and R. V. Khokhlov. Quasi-planes waves in the nonlinear acoustics of confined beams. Sov. Phys. Acoust., 15(1):35–40, 1969.