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Efficient Computation of Analytic Bases in Evans Function Analysis of Large Systems

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Abstract

In Evans function computations of the spectra of asymptotically constant-coefficient linearized operators of large systems, a problem that becomes important is the efficient computation of global analytically varying bases for invariant subspaces of the limiting coefficient matrices. In the case that the invariant subspace is spectrally separated from its complementary invariant subspace, we propose an efficient numerical implementation of a standard projection-based algorithm of Kato, for which the key step is the solution of an associated Sylvester problem. This may be recognized as the analytic cousin of a C k algorithm developed by Dieci and collaborators based on orthogonal projection rather than eigenprojection as in our case. For a one-dimensional subspace, it reduces essentially to an algorithm of Bridges, Derks and Gottwald based on path-finding and continuation methods.

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References

  1. Alexander J.C., Sachs R. (1995). Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation. Nonlinear World 2:471–507

    MATH  MathSciNet  Google Scholar 

  2. Alexander J., Gardner R., Jones C.K.R.T. (1990). A topological invariant arising in the analysis of traveling waves. J Reine Angew Math 410:167–212

    MATH  MathSciNet  Google Scholar 

  3. Allen L., Bridges T.J. (2002). Numerical exterior algebra and the compound-matrix method. Numer Math 92:197–232

    Article  MATH  MathSciNet  Google Scholar 

  4. Beyn W.-J. (1990). The numerical computation of connecting orbits in dynamical systems. IMA J Numer Anal 10:379–405

    Article  MATH  MathSciNet  Google Scholar 

  5. Bindel, D., Demmel, J., Friedman, M.J.: Continuation of invariant subspaces for large bifurcation problems. SIAM conference on applied linear algebra. Williamsburg (2003)

  6. Bindel, D., Demmel, J., Friedman, M.J., Govaerts, W.J.F., Kuznetsov, Y.A.: Bifurcation analysis of large equilibrium systems in MATLAB. In: Proceedings of the international conference on computational science ICCS 2005 (Atlanta, GA, 2005). Part I (V. S. Sunderam et al. (eds.)), Springer, Lecture Notes in Computer Science, Vol. 3514, pp.50–57 (2005)

  7. Bridges T.J., Derks G., Gottwald G. (2002). Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Physica D 172:190–216

    Article  MATH  MathSciNet  Google Scholar 

  8. Brin L. (1998). Numerical testing of the stability of viscous shock waves. Doctoral thesis, Indiana University, USA

    Google Scholar 

  9. Brin L. (2001). Numerical testing of the stability of viscous shock waves. Math Comp 70(235):1071–1088

    Article  MATH  MathSciNet  Google Scholar 

  10. Brin L., Zumbrun K. (2002). Analytically varying eigenvectors and the stability of viscous shock waves. Mat Contemp 22:19–32

    MATH  MathSciNet  Google Scholar 

  11. Chern J.L., Dieci L. (2000). Smoothness and periodicity of some matrix decompositions. SIAM J Matrix Anal Appl 22:772–792

    Article  MATH  MathSciNet  Google Scholar 

  12. Coddington E.A., Levinson N. (1955). Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York

    MATH  Google Scholar 

  13. Demmel J.W., Dieci L., Friedman M.J. (2000). Computing connecting orbits via an improved algorithm for continuing invariant subspaces. SIAM J Sci Comput 22:81–94

    Article  MATH  MathSciNet  Google Scholar 

  14. Dieci L., Eirola T. (1999). On smooth decompositions of matrices. SIAM J Matrix Anal Appl 20:800–819

    Article  MATH  MathSciNet  Google Scholar 

  15. Dieci, L., Eirola, T.: Applications of smooth orthogonal factorizations of matrices. In: Numerical methods for bifurcation problems and large-scale dynamical systems (Minneapolis, MN, 1997), Berlin Heidelberg New York: Springer 2000 IMA Vol Math Appl, Vol.119, pp. 141–162

  16. Dieci L., Friedman M.J. (2001). Continuation of invariant subspaces. Numer Lin Alg Appl 8:317–327

    Article  MATH  MathSciNet  Google Scholar 

  17. Gardner R., Zumbrun K. (1998). The Gap Lemma and geometric criteria for instability of viscous shock profiles. Comm Pure Appl Math 51:797–855

    Article  MathSciNet  Google Scholar 

  18. Golub G.H., Van Loan C.F. (1996). Matrix computations. Johns Hopkins University Press, Baltimore

    MATH  Google Scholar 

  19. Goodman J. (1986). Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch Ration Mech Anal 95:325–344

    Article  MATH  Google Scholar 

  20. Goodman, J.: Remarks on the stability of viscous shock waves. In: Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990). Philadelphia: SIAM, pp. 66–72 (1991)

  21. Kapitula T., Sandstede B. (1998). Stability of bright solitary-wave solutions to perturbed nonlinear Schrvdinger equations. Physica D 124:58–103

    Article  MATH  MathSciNet  Google Scholar 

  22. Kato T. (1985). Perturbation theory for linear operators. Springer, Berlin Heidelberg NewYork

    Google Scholar 

  23. Keller H.B. (2001). Continuation and bifurcations in scientific computation. Math Today 37:76–83

    MathSciNet  Google Scholar 

  24. Lord G.J., Peterhof D., Sandstede B., Scheel A. (2000). Numerical computation of solitary waves in infinite cylindrical domains. SIAM J Numer Anal 37:1420–1454

    Article  MATH  MathSciNet  Google Scholar 

  25. Mascia C., Zumbrun K. (2002). Pointwise Green’s function bounds and stability of relaxation shocks. Indiana Univ Math J 51:773–904

    Article  MATH  MathSciNet  Google Scholar 

  26. Mascia, C., Zumbrun, K.: Pointwise Green’s function bounds for shock profiles with degenerate viscosity. Arch Ration Mech Anal (in press)

  27. Ng B.S., Reid W.H. (1979). An initial value method for eigenvalue problems using compound matrices. J Comput Phys 30:125–136

    Article  MATH  MathSciNet  Google Scholar 

  28. Ng B.S., Reid W.H. (1979). A numerical method for linear two-point boundary value problems using compound matrices. J Comput Phys 33:70–85

    Article  MATH  MathSciNet  Google Scholar 

  29. Ng B.S., Reid W.H. (1980). On the numerical solution of the Orr-Sommerfeld problem: asymptotic initial conditions for shooting methods. J Comput Phys 38:275–293

    Article  MATH  MathSciNet  Google Scholar 

  30. Ng B.S., Reid W.H. (1985). The compound matrix method for ordinary differential systems. J Comput Phys 58:209–228

    Article  MATH  MathSciNet  Google Scholar 

  31. Sandstede, B.: Little Compton meeting on Evans function techniques (1998)

  32. Sandstede, B.: Stability of travelling waves. In: Handbook of dynamical systems II. Amsterdam: North-Holland, pp. 983–1055 (2002)

  33. Stoer J., Bulirsch R. (2002). Introduction to numerical analysis. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  34. Zumbrun K. (2004). Stability of large-amplitude shock waves of compressible Navier–Stokes equations. In: Jenssen H.K., Lyng G. (eds) Handbook of mathematical fluid dynamics, vol III. North-Holland, Amsterdam, pp. 311-533

    Google Scholar 

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Authors and Affiliations

  1. Brigham Young University, Provo, UT, 84602, USA

    Jeffrey Humpherys

  2. Department of Mathematics and Statistics, University of Surrey, Guild-ford, GU2 7XH, UK

    Björn Sandstede

  3. Indiana University, Bloomington, IN, 47405, USA

    Kevin Zumbrun

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  1. Jeffrey Humpherys
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  2. Björn Sandstede
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  3. Kevin Zumbrun
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Correspondence to Kevin Zumbrun.

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Humpherys, J., Sandstede, B. & Zumbrun, K. Efficient Computation of Analytic Bases in Evans Function Analysis of Large Systems. Numer. Math. 103, 631–642 (2006). https://doi.org/10.1007/s00211-006-0004-7

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  • DOI: https://doi.org/10.1007/s00211-006-0004-7

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