Evans function computation for the stability of travelling waves. (English) Zbl 1402.65052
Philos. Trans. R. Soc. Lond., A, Math. Phys. Eng. Sci. 376, No. 2117, Article ID 20170184, 25 p. (2018).
Summary: In recent years, the Evans function has become an important tool for the determination of stability of travelling waves. This function, a Wronskian of decaying solutions of the eigenvalue equation, is useful both analytically and computationally for the spectral analysis of the linearized operator about the wave. In particular, Evans-function computation allows one to locate any unstable eigenvalues of the linear operator (if they exist); this allows one to establish spectral stability of a given wave and identify bifurcation points (loss of stability) as model parameters vary. In this paper, we review computational aspects of the Evans function and apply it to multidimensional detonation waves.
MSC:
| 65L05 | Numerical methods for initial value problems involving ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 35C07 | Traveling wave solutions |
| 35Q30 | Navier-Stokes equations |
| 35B35 | Stability in context of PDEs |
References:
| [1] | Meshkov, EE, Instability of the interface of two gases accelerated by a shock wave, Fluid Dyn., 4, 101-104, (1969) · doi:10.1007/BF01015969 |
| [2] | Urtiew, PA; Oppenheim, AK, Experimental observations of the transition to detonation in an explosive gas, Proc. R. Soc. Lond. A, 295, 13-28, (1966) · doi:10.1098/rspa.1966.0223 |
| [3] | Torruellas, WE; Wang, Z.; Hagan, DJ; VanStryland, EW; Stegeman, GI; Torner, L.; Menyuk, CR, Observation of two-dimensional spatial solitary waves in a quadratic medium, Phys. Rev. Lett., 74, 5036-5039, (1995) · doi:10.1103/PhysRevLett.74.5036 |
| [4] | Davis, RE; Acrivos, A., Solitary internal waves in deep water, J. Fluid Mech., 29, 593-607, (1967) · Zbl 0147.46503 · doi:10.1017/S0022112067001041 |
| [5] | Rauprich, O.; Matsushita, M.; Weijer, CJ; Siegert, F.; Esipov, SE; Shapiro, JA, Periodic phenomena in proteus mirabilis swarm colony development, J. Bacteriol., 178, 6525-6538, (1996) · doi:10.1128/jb.178.22.6525-6538.1996 |
| [6] | Niemela, JJ; Ahlers, G.; Cannell, DS, Localized traveling-wave states in binary-fluid convection, Phys. Rev. Lett., 64, 1365-1368, (1990) · doi:10.1103/PhysRevLett.64.1365 |
| [7] | Durham, AC; Ridgway, EB, Control of chemotaxis in physarum polycephalum, J. Cell. Biol., 69, 218-223, (1976) · doi:10.1083/jcb.69.1.218 |
| [8] | Evans, JW, Nerve axon equations: I. Linear approximations, Indiana Univ. Math. J., 21, 877-885 · Zbl 0235.92002 · doi:10.1512/iumj.1972.21.21071 |
| [9] | Evans, JW, Nerve axon equations: II. Stability at rest, Indiana Univ. Math. J., 22, 75-90 · Zbl 0236.92010 · doi:10.1512/iumj.1973.22.22009 |
| [10] | Evans, JW, Nerve axon equations: III. Stability of the nerve impulse, Indiana Univ. Math. J., 22, 577-593 · Zbl 0245.92004 · doi:10.1512/iumj.1973.22.22048 |
| [11] | Evans, JW, Nerve axon equations: IV. The stable and the unstable impulse, Indiana Univ. Math. J., 24, 1169-1190 · Zbl 0317.92006 · doi:10.1512/iumj.1975.24.24096 |
| [12] | Zumbrun, K.; Jenssen, HK; Lyng, G., Stability of large-amplitude shock waves of compressible Navier-Stokes equations. In \(Handbook of mathematical fluid dynamics\), vol. 3 (eds S Friedlander, D Serre), pp. 311-533. Amsterdam, The Netherlands: North-Holland, (2005) |
| [13] | Sandstede, B., Stability of travelling waves. In \(Handbook of dynamical systems\), vol. 2 (ed. B Fiedler), pp. 983-1055. Amsterdam, The Netherlands: Elsevier Science, (2002) · Zbl 1056.35022 |
| [14] | Kapitula, T.; Promislow, K., With a foreword by Christopher K. R. T. Jones, Spectral and dynamical stability of nonlinear waves. Applied Mathematical Sciences, vol. 185, (2013), Springer · Zbl 1297.37001 |
| [15] | Alexander, J.; Gardner, R.; Jones, C., A topological invariant arising in the stability analysis of travelling waves, J. Reine Angew. Math., 410, 167-212, (1990) · Zbl 0705.35070 · doi:10.1515/crll.1990.410.167 |
| [16] | Humpherys, J.; Lyng, G.; Zumbrun, K., Spectral stability of ideal-gas shock layers, Arch. Ration. Mech. Anal., 194, 1029-1079, (2009) · Zbl 1422.76122 · doi:10.1007/s00205-008-0195-4 |
| [17] | Ghazaryan, A.; Humpherys, J.; Lytle, J., Spectral behavior of combustion fronts with high exothermicity, SIAM J. Appl. Math., 73, 422-437, (2013) · Zbl 1276.80005 · doi:10.1137/120864891 |
| [18] | Barker, B.; Humpherys, J.; Lyng, G.; Zumbrun, K., Viscous hyperstabilization of detonation waves in one space dimension, SIAM J. Appl. Math., 75, 885-906, (2015) · Zbl 1320.35271 · doi:10.1137/140980223 |
| [19] | Henry, D., \(Geometric theory of semilinear parabolic equations\), (1981), Springer · Zbl 0456.35001 |
| [20] | Ghazaryan, A.; Latushkin, Y.; Schecter, S., Stability of traveling waves for a class of reaction-diffusion systems that arise in chemical reaction models, SIAM J. Math. Anal., 42, 2434-2472, (2010) · Zbl 1227.35057 · doi:10.1137/100786204 |
| [21] | Alexander, JC; Sachs, R., Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation, Nonlinear World, 2, 471-507, (1995) · Zbl 0833.34046 |
| [22] | Gilbert, F.; Backus, G., Propagator matrices in elastic wave and vibration problems, Geophysics, 31, 326-332, (1966) · doi:10.1190/1.1439771 |
| [23] | Ng, BS; Reid, WH, An initial value method for eigenvalue problems using compound matrices, J. Comput. Phys., 30, 125-136, (1979) · Zbl 0408.65061 · doi:10.1016/0021-9991(79)90091-3 |
| [24] | Ng, BS; Reid, WH, A numerical method for linear two-point boundary value problems using compound matrices, J. Comput. Phys., 33, 70-85, (1979) · Zbl 0484.65053 · doi:10.1016/0021-9991(79)90028-7 |
| [25] | Ng, BS; Reid, WH, On the numerical solution of the Orr-Sommerfeld problem: asymptotic initial conditions for shooting methods, J. Comput. Phys., 38, 275-293, (1980) · Zbl 0468.76039 · doi:10.1016/0021-9991(80)90150-3 |
| [26] | Ng, BS; Reid, WH, The compound matrix method for ordinary differential systems, J. Comput. Phys., 58, 209-228, (1985) · Zbl 0576.34014 · doi:10.1016/0021-9991(85)90177-9 |
| [27] | Brin, LQ, Numerical testing of the stability of viscous shock waves. PhD thesis, Indiana University, Bloomington, (1998) |
| [28] | Brin, LQ, Numerical testing of the stability of viscous shock waves, Math. Comp., 70, 1071-1088, (2001) · Zbl 0980.65092 · doi:10.1090/S0025-5718-00-01237-0 |
| [29] | Brin, LQ; Zumbrun, K., Analytically varying eigenvectors and the stability of viscous shock waves, Mat. Contemp., 22, 19-32, (2002) · Zbl 1044.35057 |
| [30] | Bridges, TJ; Derks, G.; Gottwald, G., Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework, Physica D, 172, 190-216, (2002) · Zbl 1047.37053 · doi:10.1016/S0167-2789(02)00655-3 |
| [31] | Allen, L.; Bridges, TJ, Numerical exterior algebra and the compound matrix method, Numer. Math., 92, 197-232, (2002) · Zbl 1012.65079 · doi:10.1007/s002110100365 |
| [32] | Humpherys, J.; Zumbrun, K., An efficient shooting algorithm for Evans function calculations in large systems, Physica D, 220, 116-126, (2006) · Zbl 1101.65082 · doi:10.1016/j.physd.2006.07.003 |
| [33] | Drury, LO, Numerical solution of Orr-Sommerfeld-type equations, J. Comput. Phys., 37, 133-139, (1980) · Zbl 0448.65057 · doi:10.1016/0021-9991(80)90008-X |
| [34] | Barker, B.; Freist"e;uhler, H.; Zumbrun, K., Convex entropy, Hopf bifurcation, and viscous and inviscid shock stability, Arch. Rational Mech. Anal., 217, 309-372, (2015) · Zbl 1329.35240 · doi:10.1007/s00205-014-0838-6 |
| [35] | Humpherys, J., On the shock wave spectrum for isentropic gas dynamics with capillarity, J. Differ. Equ., 246, 2938-2957, (2009) · Zbl 1163.35454 · doi:10.1016/j.jde.2008.07.028 |
| [36] | Ledoux, V.; Malham, SJA; Thümmler, V., Grassmannian spectral shooting, Math. Comp., 79, 1585-1619, (2010) · Zbl 1196.65132 · doi:10.1090/S0025-5718-10-02323-9 |
| [37] | Gesztesy, F.; Latushkin, Y.; Zumbrun, K., Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves, J. Math. Pures Appl. (9), 90, 160-200, (2008) · Zbl 1161.47058 · doi:10.1016/j.matpur.2008.04.001 |
| [38] | Lafortune, S.; Lega, J.; Madrid, S., Instability of local deformations of an elastic rod: numerical evaluation of the Evans function, SIAM J. Appl. Math., 71, 1653-1672, (2011) · Zbl 1233.35188 · doi:10.1137/10081441X |
| [39] | Barker, B.; Nguyen, R.; Sandstede, B.; Ventura, N.; Wahl, C., Computing Evans functions numerically via boundary-value problems. (https://arxiv.org/abs/1710.02500), (2017) |
| [40] | Barker, B.; Johnson, MA; Noble, P.; Rodrigues, LM, Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation, Physica D, 258, 11-46, (2013) · Zbl 1304.35184 · doi:10.1016/j.physd.2013.04.011 |
| [41] | Barker, B.; Zumbrun, K., Numerical proof of stability of viscous shock profiles, Math. Models Methods Appl. Sci., 26, 1-19, (2016) · Zbl 1356.35170 · doi:10.1142/s0218202516500585 |
| [42] | Barker, B.; Humpherys, J.; Rudd, K.; Zumbrun, K., Stability of viscous shocks in isentropic gas dynamics, Comm. Math. Phys., 281, 231-249, (2008) · Zbl 1171.35071 · doi:10.1007/s00220-008-0487-4 |
| [43] | Coppel, WA, Dichotomies in stability theory. Lecture Notes in Mathematics, vol. 629, (1978), Springer · Zbl 0376.34001 |
| [44] | Kapitula, T.; Sandstede, B., Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations, Physica D, 124, 58-103, (1998) · Zbl 0935.35150 · doi:10.1016/S0167-2789(98)00172-9 |
| [45] | Gardner, RA; Zumbrun, K., The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math., 51, 797-855, (1998) · Zbl 0933.35136 · doi:10.1002/(SICI)1097-0312(199807)51:7<797::AID-CPA3>3.0.CO;2-1 |
| [46] | Benzoni-Gavage, S.; Serre, D.; Zumbrun, K., Alternate Evans functions and viscous shock waves, SIAM J. Math. Anal., 32, 929-962, (2001) · Zbl 0985.34075 · doi:10.1137/S0036141099361834 |
| [47] | Bader, G.; Ascher, U., A new basis implementation for a mixed order boundary value ode solver, SIAM J. Sci. Stat. Comput., 8, 483-500, (1987) · Zbl 0633.65084 · doi:10.1137/0908047 |
| [48] | Doedel, E.; Champneys, A.; Fairgrieve, T.; Kuznetsov, Y.; Sandstede, B.; Wang, X., Auto97: Continuation and bifurcation software for ordinary differential equations. (https://www.researchgate.net/publication/259563149), (2009) |
| [49] | Hale, N.; Moore, DR, A sixth-order extension to the MATLAB package bvp4c of J. Kierzenka and L. Shampine. Technical Report NA-08/04, Oxford University Computing Laboratory, (2008) |
| [50] | Humpherys, J.; Lafitte, O.; Zumbrun, K., Stability of isentropic Navier-Stokes shocks in the high-Mach number limit, Comm. Math. Phys., 293, 1-36, (2010) · Zbl 1195.35244 · doi:10.1007/s00220-009-0885-2 |
| [51] | Davey, A., An automatic orthonormalization method for solving stiff boundary value problems, J. Comput. Phys., 51, 343-356, (1983) · Zbl 0516.65065 · doi:10.1016/0021-9991(83)90098-0 |
| [52] | Dieci, L.; Russell, RD; Van Vleck, ES, Unitary integrators and applications to continuous orthonormalization techniques, SIAM J. Numer. Anal., 31, 261-281, (1994) · Zbl 0815.65096 · doi:10.1137/0731014 |
| [53] | Dieci, L.; Van Vleck, ES, Continuous orthonormalization for linear two-point boundary value problems revisited. In \(Dynamics of algorithms (Minneapolis, MN, 1997)\), pp. 69-90. IMA Vol. Math. Appl., vol. 118. New York, NY: Springer, (2000) · Zbl 0942.65084 |
| [54] | Dieci, L.; Van Vleck, ES, Orthonormal integrators based on householder and givens transformations, Future Gener. Comput. Syst., 19, 363-373, (2003) · doi:10.1016/S0167-739X(02)00163-2 |
| [55] | Dieci, L.; Van Vleck, ES, On the error in QR integration, SIAM J. Numer. Anal., 46, 1166-1189, (2008) · Zbl 1170.65068 · doi:10.1137/06067818X |
| [56] | Ascher, UM; Chin, H.; Reich, S., Stabilization of DAEs and invariant manifolds, Numer. Math., 67, 131-149, (1994) · Zbl 0791.65051 · doi:10.1007/s002110050020 |
| [57] | Higham, DJ, Time-stepping and preserving orthonormality, BIT, 37, 24-36, (1997) · Zbl 0891.65082 |
| [58] | Bindel, D.; Demmel, J.; Friedman, M., Continuation of invariant subspaces in large bifurcation problems, SIAM J. Sci. Comput., 30, 637-656, (2008) · Zbl 1160.65065 · doi:10.1137/060654219 |
| [59] | Dieci, L.; Elia, C.; Van Vleck, E., Exponential dichotomy on the real line: SVD and QR methods, J. Differ. Equ., 248, 287-308, (2010) · Zbl 1205.34062 · doi:10.1016/j.jde.2009.07.004 |
| [60] | Dieci, L.; Elia, C.; Van Vleck, E., Detecting exponential dichotomy on the real line: SVD and QR algorithms, BIT, 51, 555-579, (2011) · Zbl 1228.65123 · doi:10.1007/s10543-010-0306-0 |
| [61] | Zumbrun, K., Numerical error analysis for Evans function computations: a numerical gap lemma, centered-coordinate methods, and the unreasonable effectiveness of continuous orthogonalization. Preprint, arXiv:0904.0268, (2009) |
| [62] | Kato, T., \(Perturbation theory for linear operators\). Classics in Mathematics. Berlin, Germany: Springer. Reprint of the 1980 edition, (1995) |
| [63] | Humpherys, J.; Sandstede, B.; Zumbrun, K., Efficient computation of analytic bases in Evans function analysis of large systems, Numer. Math., 103, 631-642, (2006) · Zbl 1104.65081 · doi:10.1007/s00211-006-0004-7 |
| [64] | Zumbrun, K., A local greedy algorithm and higher-order extensions for global numerical continuation of analytically varying subspaces, Quart. Appl. Math., 68, 557-561, (2010) · Zbl 1195.65105 · doi:10.1090/S0033-569X-2010-01209-1 |
| [65] | Sattinger, DH, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22, 312-355, (1976) · Zbl 0344.35051 · doi:10.1016/0001-8708(76)90098-0 |
| [66] | Barker, B.; Humpherys, J.; Lyng, GD; Zumbrun, K., Balanced flux formulations for multidimensional Evans function computations for viscous shocks. Preprint, arxiv:1703. 02099, (2017) |
| [67] | Gubernov, V.; Mercer, GN; Sidhu, HS; Weber, RO, Evans function stability of combustion waves, SIAM J. Appl. Math., 63, 1259-1275, (2003) · Zbl 1024.35044 · doi:10.1137/S0036139901400240 |
| [68] | Barker, B., Evans function computation. Master’s thesis, Brigham Young University, Provo, (2009) |
| [69] | Bronski, JC, Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem, Physica D, 97, 376-397, (1996) · Zbl 1194.81093 · doi:10.1016/0167-2789(95)00311-8 |
| [70] | Humpherys, J.; Lytle, J., Root following in Evans function computation, SIAM J. Numer. Anal., 53, 2329-2346, (2015) · Zbl 1326.65157 · doi:10.1137/140975590 |
| [71] | Barker, B.; Humpherys, J.; Lytle, J.; Zumbrun, K., June 2015 STABLAB: a MATLAB-based numerical library for Evans function computation. Available in the github repository, http://github.com/nonlinear-waves/stablab/ |
| [72] | Courant, R.; Friedrichs, KO, \(Supersonic flow and shock waves\), (1976), Springer · Zbl 0365.76001 |
| [73] | Erpenbeck, JJ, Stability of steady-state equilibrium detonations, Phys. Fluids, 5, 604-614, (1962) · doi:10.1063/1.1706664 |
| [74] | Lee, HI; Stewart, DS, Calculation of linear detonation instability: one-dimensional instability of plane detonation, J. Fluid Mech., 216, 103-132, (1990) · Zbl 0698.76120 · doi:10.1017/S0022112090000362 |
| [75] | Powers, JM; Paolucci, S., Accurate spatial resolution estimates for reactive supersonic flow with detailed chemistry, AIAA J., 43, 1088-1099, (2005) · doi:10.2514/1.11641 |
| [76] | Majda, A., A qualitative model for dynamic combustion, SIAM J. Appl. Math., 41, 70-93, (1981) · Zbl 0472.76075 · doi:10.1137/0141006 |
| [77] | Humpherys, J.; Lyng, G.; Zumbrun, K., Stability of viscous detonations for Majda’s model, Physica D, 259, 63-80, (2013) · doi:10.1016/j.physd.2013.06.001 |
| [78] | Hendricks, J.; Humpherys, J.; Lyng, G.; Zumbrun, K., Stability of viscous weak detonation waves for Majda’s model, J. Dyn. Diff. Equ., 27, 237-260, (2015) · Zbl 1326.35199 · doi:10.1007/s10884-015-9440-3 |
| [79] | Faria, LM; Kasimov, AR; Rosales, RR, Study of a model equation in detonation theory, SIAM. J. Appl. Math., 74, 547-570, (2014) · Zbl 1304.35419 · doi:10.1137/130938232 |
| [80] | Gardner, RA, On the detonation of a combustible gas, Trans. Am. Math. Soc., 277, 431-468, (1983) · Zbl 0543.76152 · doi:10.1090/S0002-9947-1983-0694370-1 |
| [81] | Gasser, I.; Szmolyan, P., A geometric singular perturbation analysis of detonation and deflagration waves, SIAM J. Math. Anal., 24, 968-986, (1993) · Zbl 0783.76099 · doi:10.1137/0524058 |
| [82] | Lyng, G.; Zumbrun, K., One-dimensional stability of viscous strong detonation waves, Arch. Ration. Mech. Anal., 173, 213-277, (2004) · Zbl 1067.76041 · doi:10.1007/s00205-004-0317-6 |
| [83] | Lyng, G.; Zumbrun, K., A stability index for detonation waves in Majda’s model for reacting flow, Physica D, 194, 1-29, (2004) · Zbl 1061.35018 · doi:10.1016/j.physd.2004.01.036 |
| [84] | Jenssen, HK; Lyng, G.; Williams, M., Equivalence of low-frequency stability conditions for multidimensional detonations in three models of combustion, Indiana Univ. Math. J., 54, 1-64, (2005) · Zbl 1127.35046 · doi:10.1512/iumj.2005.54.2685 |
| [85] | Costanzino, N.; Jenssen, HK; Lyng, G.; Williams, M., Existence and stability of curved multidimensional detonation fronts, Indiana Univ. Math. J., 56, 1405-1461, (2007) · Zbl 1213.76112 · doi:10.1512/iumj.2007.56.2972 |
| [86] | Lyng, G.; Raoofi, M.; Texier, B.; Zumbrun, K., Pointwise Green function bounds and stability of combustion waves, J. Diff. Equ., 233, 654-698, (2007) · Zbl 1116.80017 · doi:10.1016/j.jde.2006.10.006 |
| [87] | Barker, B.; Humpherys, J.; Lyng, G.; Zumbrun, K., Euler vs. lagrange: the role of coordinates in practical Evans-function computations. (https://arxiv.org/abs/1707.01938), (2017) |
| [88] | Humpherys, J.; Lyng, G.; Zumbrun, K., Multidimensional stability of large-amplitude Navier-Stokes shocks, Arch. Ration. Mech. Anal., 226, 923-973, (2017) · Zbl 1386.35326 · doi:10.1007/s00205-017-1147-7 |
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