×

Root following in Evans function computation. (English) Zbl 1326.65157

Summary: Evans function computation is a useful tool in the study of asymptotically constant-coefficient linear operators such as those found in the investigation of traveling wave stability. The roots of this analytic function correspond to the eigenvalues of the operator, and by employing both winding number and root-finding methods, one can demonstrate spectral stability and determine the precise location of the eigenvalues up to numerical tolerance. It is often desirable to track these eigenvalues as parameters vary in a system, particularly as they may cross into the right half of the complex plane, thus signaling the onset of instability. In this paper, we present a numerical method to track these eigenvalues using continuation to follow the roots of the Evans function instead of using winding number and root-finding techniques, therefore saving time and effort. We then consider some examples.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs

Software:

GitHub; STABLAB; AUTO; COCO; COLNEW
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. Alexander, R. Gardner, and C. Jones, {\it A topological invariant arising in the stability analysis of travelling waves}, J. Reine Angew. Math., 410 (1990), pp. 167-212. · Zbl 0705.35070
[2] J. C. Alexander and R. Sachs, {\it Linear instability of solitary waves of a Boussinesq-type equation: A computer assisted computation}, Nonlinear World, 2 (1995), pp. 471-507. · Zbl 0833.34046
[3] L. Allen and T. J. Bridges, {\it Numerical exterior algebra and the compound matrix method}, Numer. Math., 92 (2002), pp. 197-232. · Zbl 1012.65079
[4] U. M. Ascher, H. Chin, and S. Reich, {\it Stabilization of DAEs and invariant manifolds}, Numer. Math., 67 (1994), pp. 131-149. · Zbl 0791.65051
[5] G. Bader and U. Ascher, {\it A new basis implementation for a mixed order boundary value ODE solver}, SIAM J. Sci. Stat. Comput., 8 (1987), pp. 483-500. · Zbl 0633.65084
[6] B. Barker, {\it Evans Function Computation}, Master’s thesis, Brigham Young University, Provo, Utah, 2009.
[7] B. Barker, J. Humpherys, G. Lyng, and K. Zumbrun, {\it Viscous hyperstabilization of detonation waves in one space dimension}, SIAM J. Appl. Math., 75 (2015), pp. 885-906. · Zbl 1320.35271
[8] B. Barker, J. Humpherys, J. Lytle, and K. Zumbrun, {\it STABLAB: A MATLAB-Based Numerical Library for Evans Function Computation}, http://github.com/nonlinear-waves/stablab/ (2015).
[9] B. Barker, J. Humpherys, K. Rudd, and K. Zumbrun, {\it Stability of viscous shocks in isentropic gas dynamics}, Comm. Math. Phys., 281 (2008), pp. 231-249. · Zbl 1171.35071
[10] W.-J. Beyn, Y. Latushkin, and J. Rottmann-Matthes, {\it Finding eigenvalues of holomorphic Fredholm operator pencils using boundary value problems and contour integrals}, Integral Equations Operator Theory, 78 (2014), pp. 155-211. · Zbl 1318.47021
[11] D. Bindel, J. Demmel, and M. Friedman, {\it Continuation of invariant subspaces in large bifurcation problems}, SIAM J. Sci. Comput., 30 (2008), pp. 637-656. · Zbl 1160.65065
[12] T. J. Bridges, G. Derks, and G. Gottwald, {\it Stability and instability of solitary waves of the fifth-order KdV equation: A numerical framework}, Phys. D, 172 (2002), pp. 190-216. · Zbl 1047.37053
[13] L. Q. Brin, {\it Numerical Testing of the Stability of Viscous Shock Waves}, Ph.D. thesis, Indiana University, Bloomington, IN, 1998. · Zbl 0980.65092
[14] L. Q. Brin, {\it Numerical testing of the stability of viscous shock waves}, Math. Comp., 70 (2001), pp. 1071-1088. · Zbl 0980.65092
[15] L. Q. Brin and K. Zumbrun, {\it Analytically varying eigenvectors and the stability of viscous shock waves}, Mat. Contemp., 22 (2002), pp. 19-32. · Zbl 1044.35057
[16] J. C. Bronski, {\it Semiclassical eigenvalue distribution of the Zakharov-Shabat eigenvalue problem}, Phys. D, 97 (1996), pp. 376-397. · Zbl 1194.81093
[17] W. A. Coppel, {\it Dichotomies in Stability Theory}, Lecture Notes in Math. 629, Springer-Verlag, Berlin, 1978. · Zbl 0376.34001
[18] H. Dankowicz and F. Schilder, {\it Recipes for Continuation}, Comput. Sci. Eng. 11, SIAM, Philadelphia, 2013. · Zbl 1277.65037
[19] A. Davey, {\it An automatic orthonormalization method for solving stiff boundary value problems}, J. Comput. Phys., 51 (1983), pp. 343-356. · Zbl 0516.65065
[20] L. Dieci, C. Elia, and E. Van Vleck, {\it Exponential dichotomy on the real line: SVD and QR methods}, J. Differential Equations, 248 (2010), pp. 287-308. · Zbl 1205.34062
[21] L. Dieci, C. Elia, and E. Van Vleck, {\it Detecting exponential dichotomy on the real line: SVD and QR algorithms}, BIT, 51 (2011), pp. 555-579. · Zbl 1228.65123
[22] L. Dieci, R. D. Russell, and E. S. Van Vleck, {\it Unitary integrators and applications to continuous orthonormalization techniques}, SIAM J. Numer. Anal., 31 (1994), pp. 261-281. · Zbl 0815.65096
[23] L. Dieci and E. S. Van Vleck, {\it Continuous Orthonormalization for Linear Two-Point Boundary Value Problems Revisited}, in Dynamics of Algorithms (Minneapolis, MN, 1997), IMA Vol. Math. Appl. 118, Springer, New York, 2000, pp. 69-90. · Zbl 0942.65084
[24] L. Dieci and E. S. Van Vleck, {\it Orthonormal integrators based on Householder and Givens transformations}, Future Gener. Comput. Syst., 19 (2003), pp. 363-373.
[25] L. Dieci and E. S. Van Vleck, {\it On the error in QR integration}, SIAM J. Numer. Anal., 46 (2008), pp. 1166-1189. · Zbl 1170.65068
[26] E. Doedel, A. Champneys, T. Fairgrieve, Y. Kuznetsov, B. Sandstede, and X. Wang, {\it Auto97: Continuation and Bifurcation Software for Ordinary Differential Equations}, www.dam.brown.edu/people/sansted/publications/auto97.pdf (1998).
[27] J. R. Dormand and P. J. Prince, {\it A family of embedded Runge-Kutta formulae}, J. Comput. Appl. Math., 6 (1980), pp. 19-26. · Zbl 0448.65045
[28] L. O’C. Drury, {\it Numerical solution of Orr-Sommerfeld-type equations}, J. Comput. Phys., 37 (1980), pp. 133-139. · Zbl 0448.65057
[29] F. Gesztesy, Y. Latushkin, and K. Zumbrun, {\it Derivatives of (modified) Fredholm determinants and stability of standing and traveling waves}, J. Math. Pures Appl. (9), 90 (2008), pp. 160-200. · Zbl 1161.47058
[30] A. Ghazaryan, J. Humpherys, and J. Lytle, {\it Spectral behavior of combustion fronts with high exothermicity}, SIAM J. Appl. Math., 73 (2013), pp. 422-437. · Zbl 1276.80005
[31] A. Ghazaryan, Y. Latushkin, and S. Schecter, {\it Stability of traveling waves for a class of reaction-diffusion systems that arise in chemical reaction models}, SIAM J. Math. Anal., 42 (2010), pp. 2434-2472. · Zbl 1227.35057
[32] F. Gilbert and G.E. Backus, {\it Propagator matrices in elastic wave and vibration problems}, Geophysics, 31 (1966), pp. 326-332.
[33] N. Hale and D. R. Moore, {\it A Sixth-Order Extension to the MATLAB Package BVP \(4C\) of J. Kierzenka and L. Shampine}, Technical report NA-08/04, Oxford University Computing Laboratory, Oxford, May 2008.
[34] D. Henry, {\it Geometric Theory of Semilinear Parabolic Equations}, Springer-Verlag, Berlin, 1981. · Zbl 0456.35001
[35] D. J. Higham, {\it Time-stepping and preserving orthonormality}, BIT, 37 (1997), pp. 24-36. · Zbl 0891.65082
[36] J. Humpherys, O. Lafitte, and K. Zumbrun, {\it Stability of isentropic Navier-Stokes shocks in the high-Mach number limit}, Comm. Math. Phys., 293 (2010), pp. 1-36. · Zbl 1195.35244
[37] J. Humpherys, G. Lyng, and K. Zumbrun, {\it Spectral stability of ideal-gas shock layers}, Arch. Ration. Mech. Anal., 194 (2009), pp. 1029-1079. · Zbl 1422.76122
[38] J. Humpherys, B. Sandstede, and K. Zumbrun, {\it Efficient computation of analytic bases in Evans function analysis of large systems}, Numer. Math., 103 (2006), pp. 631-642. · Zbl 1104.65081
[39] J. Humpherys and K. Zumbrun, {\it An efficient shooting algorithm for Evans function calculations in large systems}, Phys. D, 220 (2006), pp. 116-126. · Zbl 1101.65082
[40] T. Kapitula and K. Promislow, {\it Spectral and Dynamical Stability of Nonlinear Waves}, Appl. Math. Sci. 185, Springer, New York, 2013. · Zbl 1297.37001
[41] T. Kapitula and B. Sandstede, {\it Instability mechanism for bright solitary-wave solutions to the cubic-quintic Ginzburg-Landau equation}, J. Opt. Soc. Amer. B Opt. Phys., 15 (1998), pp. 2757-2762.
[42] T. Kapitula and B. Sandstede, {\it Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations}, Phys. D, 124 (1998), pp. 58-103. · Zbl 0935.35150
[43] T. Kato, {\it Perturbation Theory for Linear Operators}, Class. Math., Springer-Verlag, Berlin, 1995; reprint of the 1980 edition. · Zbl 0836.47009
[44] V. Ledoux, S. J. A. Malham, and V. Thümmler, {\it Grassmannian spectral shooting}, Math. Comp., 79 (2010), pp. 1585-1619. · Zbl 1196.65132
[45] G. N. Mercer, H. S. Sidhu, R. O. Weber, and V. Gubernov, {\it Evans function stability of combustion waves}, SIAM J. Appl. Math., 63 (2003), pp. 1259-1275. · Zbl 1024.35044
[46] B. S. Ng and W. H. Reid, {\it An initial value method for eigenvalue problems using compound matrices}, J. Comput. Phys., 30 (1979), pp. 125-136. · Zbl 0408.65061
[47] B. S. Ng and W. H. Reid, {\it A numerical method for linear two-point boundary value problems using compound matrices}, J. Comput. Phys., 33 (1979), pp. 70-85. · Zbl 0484.65053
[48] B. S. Ng and W. H. Reid, {\it On the numerical solution of the Orr-Sommerfeld problem: Asymptotic initial conditions for shooting methods}, J. Comput. Phys., 38 (1980), pp. 275-293. · Zbl 0468.76039
[49] B. S. Ng and W. H. Reid, {\it The compound matrix method for ordinary differential systems}, J. Comput. Phys., 58 (1985), pp. 209-228. · Zbl 0576.34014
[50] K. J. Palmer, {\it Exponential dichotomies and transversal homoclinic points}, J. Differential Equations, 55 (1984), pp. 225-256. · Zbl 0508.58035
[51] R. L. Pego and M. I. Weinstein, {\it Eigenvalues, and instabilities of solitary waves}, R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci., 340 (1992), pp. 47-94. · Zbl 0776.35065
[52] K. Zumbrun, {\it 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.
[53] K. Zumbrun, {\it A local greedy algorithm and higher-order extensions for global numerical continuation of analytically varying subspaces}, Quart. Appl. Math., 68 (2010), pp. 557-561. · Zbl 1195.65105
[54] K. Zumbrun and P. Howard, {\it Pointwise semigroup methods and stability of viscous shock waves}, Indiana Univ. Math. J., 47 (1998), pp. 741-871. · Zbl 0928.35018
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.