Skip to main content
SpringerLink
Log in

Numerical Evaluation of the Evans Function by Magnus Integration

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

Abstract

We use Magnus methods to compute the Evans function for spectral problems as arise when determining the linear stability of travelling wave solutions to reaction-diffusion and related partial differential equations. In a typical application scenario, we need to repeatedly sample the solution to a system of linear non-autonomous ordinary differential equations for different values of one or more parameters as we detect and locate the zeros of the Evans function in the right half of the complex plane.

In this situation, a substantial portion of the computational effort—the numerical evaluation of the iterated integrals which appear in the Magnus series—can be performed independent of the parameters and hence needs to be done only once. More importantly, for any given tolerance Magnus integrators possess lower bounds on the step size which are uniform across large regions of parameter space and which can be estimated a priori. We demonstrate, analytically as well as through numerical experiment, that these features render Magnus integrators extremely robust and, depending on the regime of interest, efficient in comparison with standard ODE solvers.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Google Scholar 

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

  3. H. F. Baker, On the integration of linear differential equations, Proc. Lond. Math. Soc., 35 (1903), pp. 333–378.

  4. N. J. Balmforth, R. V. Craster and S. J. A. Malham, Unsteady fronts in an autocatalytic system, Proc. R. Soc. Lond. A, 455 (1999), pp. 1401–1433.

  5. I. Bialynicki-Birula, B. Mielnik and J. Plebanski, Explicit solution of the continuous Baker-Campbell-Hausdorff problem and a new expression for the phase operator, Ann. Phys., 51 (1969), pp. 187–200.

    Google Scholar 

  6. J. Billingham and D. Needham, The development of travelling waves in quadratic and cubic autocatalysis with unequal diffusion rates, I and II, Philos. Trans. R. Soc. Lond. A, 334 (1991), pp. 1–124, and 336 (1991), pp. 497–539.

  7. S. Blanes, F. Casas and J. Ros, Optimized geometric integrators of high order for linear differential equations, BIT, 42 (2002), pp. 262–284.

  8. S. Blanes, F. Casas, J. A. Oteo and J. Ros, Magnus and Fer expansions for matrix differential equations: the convergence problems, J. Phys A: Math. Gen., 31 (1998), pp. 259–268.

    Google Scholar 

  9. B. Chanane, Fleiss series approach to the computation of the eigenvalues of fourth–order Sturm–Liouville problems, Appl. Math. Lett., 15 (2002), pp. 459–463.

    Google Scholar 

  10. K. T. Chen, Integration of paths, geometric invariants and a generalized Baker–Hausdorff formula, Ann. Math., 65(1) (1957), pp. 163–178.

    Google Scholar 

  11. S. Coombes and M. R. Owen, Evans functions for integral neural field equations with heaviside firing rate function, SIAM J. Appl. Dyn. Syst., 3 (2004), pp. 574–600.

    Google Scholar 

  12. I. Degani and J. Schiff, Right Correction Magnus Series Approach for Integration of Linear Ordinary Differential Equations with Highly Oscillatory Solution, Technical report MCS03-04, Mathematics and Computer Science, Weizmann Institute of Science, 2003.

  13. F. J. Dyson, The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev., 75(3) (1949), pp. 486–502.

  14. J. W. Evans, Nerve axon equations, IV: The stable and unstable impulse. Indiana Univ. Math. J., 24 (1975), pp. 1169–1190.

    Google Scholar 

  15. L. Greenberg and M. Marletta, Numerical solution of non-self-adjoint Sturm-Liouville problems and related systems, SIAM J. Numer. Anal., 38(6) (2001), pp. 1800–1845.

  16. E. Hairer, S. P. Nørsett and G. Wanner, Solving ordinary differential equations I, Springer-Verlag, Springer Series in Computational Mathematics 8, 1987.

  17. E. Hairer and G. Wanner, Solving ordinary differential equations II, Springer-Verlag, Springer Series in Computational Mathematics 14, 1991.

  18. D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, Lecture Notes in Mathematics 840, 1981.

  19. M. Hochbruck and C. Lubich, On Magnus integrators for time-dependent Schrödinger equations, SIAM J. Numer. Anal., 41 (2003), pp. 945–963.

    Google Scholar 

  20. A. Iserles, Think globally, act locally: solving highly-oscillatory ordinary differential equations, Appl. Numer. Math., 43 (2002), pp. 145–160.

    Google Scholar 

  21. A. Iserles, On the method of Neumann series for highly oscillatory equations, BIT, 44 (2004), pp. 473–488.

  22. A. Iserles, A. Marthinsen and S. P. Nørsett, On the implementation of the method of Magnus series for linear differential equations, BIT, 39 (1999), pp. 281–304.

  23. A. Iserles, H. Z. Munthe-Kaas, S. P. Nørsett and A. Zanna, Lie-group methods, Acta Numer., (2000), pp. 215–365.

  24. A. Iserles and S. P. Nørsett, On the solution of linear differential equations in Lie groups, Philos. Trans. R. Soc. Lond. A, 357 (1999), pp. 983–1019.

    Google Scholar 

  25. A. Iserles and A. Zanna, Efficient computation of the matrix exponential by generalized polar decompositions, Technical report 2002/NA09, DAMTP, University of Cambridge, 2002.

  26. L. Jódar and M. Marletta, Solving ODEs arising from non-selfadjoint Hamiltonian eigenproblems, Adv. Comput. Math., 13 (2000), pp. 231–256.

  27. T. Kapitula, The Evans function and generalized Melnikov integrals, SIAM J. Math. Anal., 30 (1998), pp. 273–297.

  28. D. E. Knuth, The art of Computer Programming, Vol. 2: Seminumerical Algorithms, Second Edition, Addison-Wesley Publishing Company, 1981.

  29. W. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math., 7 (1954), pp. 649–673.

  30. P.-C. Moan, Efficient approximation of Sturm-Liouville problems using Lie-group methods, Technical report 1998/NA11, DAMTP, University of Cambridge, 1998.

  31. P.-C. Moan, On backward error analysis and Nekhoroshev stability in the numerical analysis of conservative systems of ODEs, PhD Thesis, University of Cambridge, 2002.

  32. C. Moler and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Rev., 45 (2003), pp. 3–49.

    Google Scholar 

  33. H. Munthe-Kaas and B. Owren, Computations in a free Lie algebra, Philos. Trans. R. Soc. Lond. A, 357 (1999), pp. 957–981.

    Google Scholar 

  34. G. Peano, Intégration par séries des équations différentielles linéaires, Math. Ann., XXXII (1888), pp. 450–457.

  35. R. L. Pego and M. I. Weinstein, Eigenvalues, and instabilities of solitary waves, Philos. Trans. R. Soc. Lond. A, 340 (1992), pp. 47–94.

  36. J. D. Pryce, Numerical solution of Sturm–Liouville problems, Monographs on Numerical Analysis, Oxford Science Publications, Clarendon Press, 1993.

  37. M. Reed and B. Simon, Methods of modern mathematical physics II: Fourier analysis, self–adjointness, Academic Press, 1975.

  38. B. Sandstede, Stability of travelling waves, in Handbook of Dynamical Systems II, B. Fiedler, ed., Elsevier, 2002, pp. 983–1055.

  39. A. M. Stuart and A. R. Humphries, Dynamical systems and numerical analysis, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 1996.

  40. D. Terman, Stability of planar wave solutions to a combustion model, SIAM J. Math. Anal., 21 (1990), pp. 1139–1171.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Oxford Brookes University, Wheatley, OX33 1HX, UK

    Nairo D. Aparicio

  2. Department of Mathematics, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Simon J. A. Malham

  3. School of Engineering and Science, International University Bremen, 28759, Bremen, Germany

    Marcel Oliver

Authors
  1. Nairo D. Aparicio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon J. A. Malham
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Marcel Oliver
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Nairo D. Aparicio, Simon J. A. Malham or Marcel Oliver.

Additional information

AMS subject classification (2000)

65F20

Rights and permissions

Reprints and permissions

About this article

Cite this article

Aparicio, N., Malham, S. & Oliver, M. Numerical Evaluation of the Evans Function by Magnus Integration. Bit Numer Math 45, 219–258 (2005). https://doi.org/10.1007/s10543-005-0001-8

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10543-005-0001-8

Key words

Search

Navigation