A local greedy algorithm and higher-order extensions for global numerical continuation of analytically varying subspaces
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In this article, the authors 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.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.read more
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Journal ArticleDOI
One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics
TL;DR: In this article, a combination of asymptotic ODE estimates and numerical Evans function computations was used to study the stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters (shock amplitude, strength of imposed magnetic field, viscosity, magnetic permeability, and electrical resistivity) for a γ-law gas with γ ∈ [ 1, 3 ].
Posted Content
Numerical error analysis for Evans function computations: a numerical gap lemma, centered-coordinate methods, and the unreasonable effectiveness of continuous orthogonalization
TL;DR: In this article, error analysis of the Drury method is performed in the context of numerical approximations of the Evans function, showing that it is in fact (neutrally) stable when used to approximate an unstable subspace.
Journal ArticleDOI
Existence and Stability of Viscoelastic Shock Profiles
TL;DR: In this article, the existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani were investigated.
Journal ArticleDOI
Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity
TL;DR: In this paper, a global analysis of all possible viscous shock profiles was carried out for the two-dimensional Navier-Stokes equations of isentropic magnetohydrodynamics (MHD).
Book ChapterDOI
Stability and Dynamics of Viscous Shock Waves
TL;DR: In this paper, the stability, dynamics, and bifurcation of viscous shock waves and related solutions of nonlinear pde were examined from a classical dynamical systems point of view.
References
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Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
Journal ArticleDOI
The gap lemma and geometric criteria for instability of viscous shock profiles
Robert A. Gardner,Kevin Zumbrun +1 more
TL;DR: In this article, the gap lemma was shown to be a necessary condition for the stability of viscous wave profiles, which is defined in terms of the sign of a certain Melnikov integral of the associated viscous profile.
Journal ArticleDOI
Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework
TL;DR: In this article, a robust shooting algorithm based on exterior algebra spaces is introduced to solve the spectral problem associated with the linearization about solitary waves of the generalized fifth-order KdV equation.
Journal ArticleDOI
Numerical testing of the stability of viscous shock waves
TL;DR: A new theoretical Evans function condition is used as the basis of a numerical test of viscous shock wave stability, and the need to incorporate features from the analytic Evans function theory for purposes of numerical stability is found.
Journal ArticleDOI
An efficient shooting algorithm for Evans function calculations in large systems
Jeffrey Humpherys,Kevin Zumbrun +1 more
TL;DR: In this article, Brin and Zumbrun introduced a simple polar coordinate algorithm representing pure (monomial) products as scalar multiples of orthonormal bases, for which the angular equation is a numerically optimized version of the continuous orthogonalization method of Drury-Davey.
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