Computing stability of multi-dimensional travelling waves

We present a numerical method for computing the pure-point spectrum associated with the linear stability of multi-dimensional travelling fronts to parabolic nonlinear systems. Our method is based on the Evans function shooting approach. Transverse to the direction of propagation we project the spectral equations onto a finite Fourier basis. This generates a large, linear, one-dimensional system of equations for the longitudinal Fourier coefficients. We construct the stable and unstable solution subspaces associated with the longitudinal far-field zero boundary conditions, retaining only the information required for matching, by integrating the Riccati equations associated with the underlying Grassmannian manifolds. The Evans function is then the matching condition measuring the linear dependence of the stable and unstable subspaces and thus determines eigenvalues. As a model application, we study the stability of two-dimensional wrinkled front solutions to a cubic autocatalysis model system. We compare our shooting approach with the continuous orthogonalization method of Humpherys and Zumbrun. We then also compare these with standard projection methods that directly project the spectral problem onto a finite multi-dimensional basis satisfying the boundary conditions.Comment: 23 pages, 9 figures (some in colour). v2: added details and other changes to presentation after referees' comments, now 26 page

Numerical analysis of the method of freezing traveling waves

Thümmler V. Numerical analysis of the method of freezing traveling waves. Bielefeld (Germany): Bielefeld University; 2005.In der vorliegenden Arbeit werden spezielle Lösungen von parabolischen partiellen Differentialgleichungen (PDE) u_t = A u_xx + f(u, u_x), nämlich wandernde Wellen der Form u(x,t) = w(x-c t), untersucht. Dabei ist w das Wellenprofil und c die Geschwindigkeit. Das Paar (w,c) lässt sich als Gleichgewicht einer partiell differentiell algebraischen Gleichung (PDAE) auffassen, die aus PDE entsteht, indem man den Ansatz u(x,t) = v(x-g(t),t) in PDE einsetzt und eine zusätzliche Phasenbedingung einführt. Durch Diskretisierung von PDAE mit dem finiten Differenzenverfahren auf einem endlichen Gitter erhält man eine differentiell algebraische Gleichung (DAE). In der Arbeit werden sowohl der Effekt der Transformation PDE -> PDAE (das "Einfrieren der Welle") als auch der Diskretisierung PDAE -> DAE auf die Existenz und Stabilität von wandernden Wellen, bzw. allgemeiner von relativen Gleichgewichten, untersucht. Eines der Hauptergebnisse ist der Nachweis, dass auch DAE (unter gewissen Voraussetzungen an die Randbedingungen) ein der eingefrorenen Welle (w,c) entsprechendes Gleichgewicht besitzt, welches die Stabilitätseigenschaften der wandernden Welle erbt.This thesis deals with special solutions of parabolic partial differential equations (PDE) u_t = A u_xx + f(u, u_x), namely traveling waves of the form u(x,t) = w(x-c t). Here w denotes the profile of the wave and c its velocity. The pair (w,c) is an equilibrium of a partial differential algebraic equation (PDAE) which is constructed by inserting the ansatz u(x,t) = v(x-g(t),t) in PDE and adding an additional phase condition. By discretization with finite differences on a finite grid, one obtains a differential algebraic equation (DAE). In the thesis the effect of the transformation PDE -> PDAE (the 'freezing of the wave') and of the discretization PDAE -> DAE to existence and stability of traveling wave solutions, or more general, of relative equilibria, is analyzed. One of the main results is the proof of the existence of an equilibrium of DAE which corresponds to (w,c) and which inherits the stability properties of the traveling wave

Freezing solutions of equivariant evolution equations

Beyn W-J, Thümmler V. Freezing solutions of equivariant evolution equations. SIAM Journal on Applied Dynamical Systems. 2004;3(2):85-116.In this paper we develop numerical methods for integrating general evolution equations u(t) = F(u), u(0) = u(0), where F is defined on a dense subspace of some Banach space (generally infinite-dimensional) and is equivariant with respect to the action of a finite-dimensional (not necessarily compact) Lie group. Such equations typically arise from autonomous PDEs on unbounded domains that are invariant under the action of the Euclidean group or one of its subgroups. In our approach we write the solution u(t) as a composition of the action of a time-dependent group element with a "frozen solution" in the given Banach space. We keep the frozen solution as constant as possible by introducing a set of algebraic constraints (phase conditions), the number of which is given by the dimension of the Lie group. The resulting PDAE (partial differential algebraic equation) is then solved by combining classical numerical methods, such as restriction to a bounded domain with asymptotic boundary conditions, half-explicit Euler methods in time, and finite differences in space. We provide applications to reaction-diffusion systems that have traveling wave or spiral solutions in one and two space dimensions

Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems

Beyn W-J, Thümmler V. Continuation of Invariant Subspaces for Parameterized Quadratic Eigenvalue Problems. SIAM Journal on Matrix Analysis and Applications. 2009;31(3):1361-1381.We consider quadratic eigenvalue problems with large and sparse matrices depending on a parameter. Problems of this type occur, for example, in the stability analysis of spatially discretized and parameterized nonlinear wave equations. The aim of the paper is to present and analyze a continuation method for invariant subspaces that belong to a group of eigenvalues, the number of which is much smaller than the dimension of the system. The continuation method is of predictor-corrector type, similar to the approach for the linear eigenvalue problem in [Beyn, Kless, and Thummler, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001], but we avoid linearizing the problem, which will double the dimension and change the sparsity pattern. The matrix equations that occur in the predictor and the corrector step are solved by a bordered version of the Bartels-Stewart algorithm. Furthermore, we set up an update procedure that handles the transition from real to complex conjugate eigenvalues, which occurs when eigenvalues from inside the continued cluster collide with eigenvalues from outside. The method is demonstrated on several numerical examples: a homotopy between random matrices, a fluid conveying pipe problem, and a traveling wave of a damped wave equation

Freezing multipulses and multifronts

Beyn W-J, Selle S, Thümmler V. Freezing multipulses and multifronts. SIAM Journal on Applied Dynamical Systems. 2008;7(2):577-608.We consider nonlinear time dependent reaction diffusion systems in one space dimension that exhibit multiple pulses or multiple fronts. In an earlier paper two of the authors developed the freezing method that allows us to compute a moving coordinate frame in which, for example, a traveling wave becomes stationary. In this paper we extend the method to handle multifronts and multipulses traveling at different speeds. The solution of the Cauchy problem is decomposed into a finite number of single waves, each of which has its own moving coordinate system. The single solutions satisfy a system of partial differential algebraic equations coupled by nonlinear and nonlocal terms. Applications are provided to the Nagumo and the FitzHugh-Nagumo systems. We justify the method by showing that finitely many traveling waves, when patched together in an appropriate way, solve the coupled system in an asymptotic sense. The method is generalized to equivariant evolution equations and is illustrated by the complex Ginzburg-Landau equation

GRASSMANNIAN SPECTRAL SHOOTING

We present a new numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. In the context of the Evans function shooting and matching approach, all the relevant information is carried by the flow projected onto the underlying Grassmann manifold. We show how to numerically construct this projected flow in a stable and robust manner. In particular, the method avoids representation singularities by, in practice, choosing the best coordinate patch representation for the flow as it evolves. The method is analytic in the spectral parameter and of complexity bounded by the order of the spectral problem cubed. For large systems it represents a competitive method to those recently developed that are based on continuous orthogonalization. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and the Ekman boundary layer

Computing the Evans function using Grassmannians

We present a numerical method for computing the pure-point spectrum associated with the linear stability of coherent structures. Our method is based on the Evans function shooting and matching approach. The Grassmann representatives for the stable and unstable manifolds of the spectral problem suffice to construct the Evans function. Our idea is to fix a coordinate patch for the Grassmann representatives of each manifold and numerically compute in that representation. We are thus required to solve a nonlinear Riccati differential equation for each manifold. In practice the method is stable, robust, analytic in the spectral parameter and of complexity bounded by the order of the spectral problem. For large systems it represents a competitive method to that proposed by Humpherys and Zumbrun [21]. We demonstrate this by comparing the two methods in three applications: Boussinesq solitary waves, autocatalytic travelling waves and Ekman boundary layer