Abstract. In this paper we develop numerical methods for integrating general evolution equations ut = F (u), u(0) = u0, where F is defined on a dense subspace of some Banach space (generally infinitedimensional) 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.

Abstract. 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.

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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 Thümmler, 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.