A local greedy algorithm and higher-order extensions for global numerical continuation of analytically varying subspaces
Author:
Kevin Zumbrun
Journal:
Quart. Appl. Math. 68 (2010), 557-561
MSC (2000):
Primary 65L99
DOI:
https://doi.org/10.1090/S0033-569X-2010-01209-1
Published electronically:
May 27, 2010
MathSciNet review:
2676976
Full-text PDF Free Access
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Abstract: We present a family of numerical implementations of Kato’s ODE propagating global bases of analytically varying invariant subspaces of which the first-order version is a surprisingly simple “greedy algorithm” that is both stable and easy to program and the second-order version a relaxation of a first-order scheme of Brin and Zumbrun. The method has application to numerical Evans function computations used to assess stability of traveling-wave solutions of time-evolutionary PDE.
References
- Blake Barker, Jeffrey Humpherys, Keith Rudd, and Kevin Zumbrun, Stability of viscous shocks in isentropic gas dynamics, Comm. Math. Phys. 281 (2008), no. 1, 231–249. MR 2403609, DOI https://doi.org/10.1007/s00220-008-0487-4
- Thomas J. Bridges, Gianne Derks, and Georg Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework, Phys. D 172 (2002), no. 1-4, 190–216. MR 1946769, DOI https://doi.org/10.1016/S0167-2789%2802%2900655-3
- Leon Q. Brin, Numerical testing of the stability of viscous shock waves, Math. Comp. 70 (2001), no. 235, 1071–1088. MR 1710652, DOI https://doi.org/10.1090/S0025-5718-00-01237-0
- Leon Q. Brin and Kevin Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves, Mat. Contemp. 22 (2002), 19–32. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). MR 1965784
- Nicola Costanzino, Jeffrey Humpherys, Toan Nguyen, and Kevin Zumbrun, Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 537–587. MR 2505363, DOI https://doi.org/10.1007/s00205-008-0153-1
- Robert A. Gardner and Kevin Zumbrun, The gap lemma and geometric criteria for instability of viscous shock profiles, Comm. Pure Appl. Math. 51 (1998), no. 7, 797–855. MR 1617251, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199807%2951%3A7%3C797%3A%3AAID-CPA3%3E3.0.CO%3B2-1
- J. Humpherys, O. Lafitte, and K. Zumbrun, Stability of isentropic viscous shock profiles in the high-mach number limit, to appear in Comm. Math. Phys.
- J. Humpherys, G. Lyng, and K. Zumbrun, Spectral stability of ideal-gas shock layers, Arch. Ration. Mech. Anal. 194 (2009), no. 3, 1029–1079.
- J. Humpherys, G. Lyng, and K. Zumbrun, Multidimensional spectral stability of large-amplitude Navier-Stokes shocks, in preparation.
- Jeffrey Humpherys, Björn Sandstede, and Kevin Zumbrun, Efficient computation of analytic bases in Evans function analysis of large systems, Numer. Math. 103 (2006), no. 4, 631–642. MR 2221065, DOI https://doi.org/10.1007/s00211-006-0004-7
- Jeffrey Humpherys and Kevin Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Phys. D 220 (2006), no. 2, 116–126. MR 2253406, DOI https://doi.org/10.1016/j.physd.2006.07.003
- T. Kato, Perturbation theory for linear operators. Springer–Verlag, Berlin, Heidelberg (1985).
References
- B. Barker, J. Humpherys, K. Rudd, and K. Zumbrun, Stability of viscous shocks in isentropic gas dynamics, Comm. Math. Phys. 281 (2008), no. 1, 231–249. MR 2403609 (2009c:35286)
- T.J. Bridges, G. Derks, and G. Gottwald, Stability and instability of solitary waves of the fifth- order KdV equation: A numerical framework, Phys. D, 172(1-4):190–216, 2002. MR 1946769 (2004i:37148)
- L.Q. Brin, Numerical testing of the stability of viscous shock waves, Math. Comp., 70(235):1071–1088, 2001. MR 1710652 (2001j:65118)
- L.Q. Brin and K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves, Mat. Contemp., 22:19–32, 2002, Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). MR 1965784 (2004c:15012)
- N. Costanzino, J. Humpherys, T. Nguyen, and K. Zumbrun, Spectral stability of noncharacteristic boundary layers of isentropic Navier–Stokes equations, Arch. Ration. Mech. Anal. 192 (2009), no. 3, 537–587. MR 2505363
- R. Gardner and K. Zumbrun, The gap lemma and geometric criteria instability of viscous shock profiles, CPAM 51. 1998, 797-855. MR 1617251 (99c:35152)
- J. Humpherys, O. Lafitte, and K. Zumbrun, Stability of isentropic viscous shock profiles in the high-mach number limit, to appear in Comm. Math. Phys.
- J. Humpherys, G. Lyng, and K. Zumbrun, Spectral stability of ideal-gas shock layers, Arch. Ration. Mech. Anal. 194 (2009), no. 3, 1029–1079.
- J. Humpherys, G. Lyng, and K. Zumbrun, Multidimensional spectral stability of large-amplitude Navier-Stokes shocks, in preparation.
- J. Humpherys, B. Sandstede, and K. Zumbrun, Efficient computation of analytic bases in Evans function analysis of large systems, Numer. Math. 103 (2006), no. 4, 631–642. MR 2221065 (2007a:65028)
- J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Physica D, 220(2):116–126, 2006. MR 2253406 (2007e:35006)
- T. Kato, Perturbation theory for linear operators. Springer–Verlag, Berlin, Heidelberg (1985).
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Additional Information
Kevin Zumbrun
Affiliation:
Department of Mathematics, 223 Rawles Hall, Indiana University, Bloomington, Indiana 47405
MR Author ID:
330192
Email:
kzumbrun@indiana.edu
Received by editor(s):
December 1, 2008
Published electronically:
May 27, 2010
Additional Notes:
This research was partially supported under NSF grants number DMS-0300487, DMS-0505780, and DMS-0801745.
Article copyright:
© Copyright 2010
Brown University