Wolf-Jürgen Beyn

Wolf-Jürgen Beyn

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least $k$ column vectors, where $k$ is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension $k$. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where $k$ is much smaller than the matrix dimension. We also give an extension of the method to the case where $k$ is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour.

Wolf-Jürgen Beyn, Denny Otten, Jens Rottmann-Matthes

In this paper we investigate the implementation of the so-called freezing method for second order wave equations in one and several space dimensions. The method converts the given PDE into a partial differential algebraic equation which is then solved numerically. The reformulation aims at separating the motion of a solution into a co-moving frame and a profile which varies as little as possible. Numerical examples demonstrate the feasability of this approach for semilinear wave equations with sufficient damping. We treat the case of a traveling wave in one space dimension and of a rotating wave in two space dimensions. In addition, we investigate in arbitrary space dimensions the point spectrum and the essential spectrum of operators obtained by linearizing about the profile, and we indicate the consequences for the nonlinear stability of the wave.

Wolf-Jürgen Beyn, Denny Otten, Jens Rottmann-Matthes

The topic of this paper are nonlinear traveling waves occuring in a system of damped waves equations in one space variable. We extend the freezing method from first to second order equations in time. When applied to a Cauchy problem, this method generates a comoving frame in which the solution becomes stationary. In addition it generates an algebraic variable which converges to the speed of the wave, provided the original wave satisfies certain spectral conditions and initial perturbations are sufficiently small. We develop a rigorous theory for this effect by recourse to some recent nonlinear stability results for waves in first order hyperbolic systems. Numerical computations illustrate the theory for examples of Nagumo and FitzHugh-Nagumo type.

Wolf-Jürgen Beyn, Denny Otten

In this paper we study spectra and Fredholm properties of Ornstein-Uhlenbeck operators $$\mathcal{L}v(x)=A\triangle v(x)+\langle Sx,\nabla v(x)\rangle+Df(v_{\star}(x))v(x),\,x\in\mathbb{R}^d,\,d\geqslant 2$$ where $v_{\star}:\mathbb{R}^d\rightarrow\mathbb{R}^m$ is a rotating wave profile with $v_{\star}(x)\to v_{\infty}\in\mathbb{R}^m$ as $|x|\to\infty$, $f:\mathbb{R}^m\rightarrow\mathbb{R}^m$ is smooth, $A\in\mathbb{R}^{m,m}$ has eigenvalues with positive real parts and commutes with the limit matrix $Df(v_{\infty})$. The matrix $S\in\mathbb{R}^{d,d}$ is assumed to be skew-symmetric with eigenvalues $(λ_1,\ldots,λ_d)=(\pm iσ_1,\ldots,\pm i σ_k,0,\ldots,0)$. The spectra of these linearized operators are crucial for the nonlinear stability of rotating waves in reaction diffusion systems. We prove under suitable conditions that every $λ\in\mathbb{C}$ satisfying the dispersion relation $$\det\Big(λI_m + η^2 A - Df(v_{\infty}) + i\langle n,σ\rangle I_m\Big)=0\quad\text{for some $η\in\mathbb{R}$ and $n\in\mathbb{Z}^k$}$$ belongs to the essential spectrum $σ_{\mathrm{ess}}(\mathcal{L})$ in $L^p$. For values $\mathrm{Re}\,λ$ to the right of the spectral bound for $Df(v_{\infty})$ we show that the operator $λI-\mathcal{L}$ is Fredholm of index $0$, solve the identification problem for the adjoint operator $(λI-\mathcal{L})^*$, and formulate the Fredholm alternative. Moreover, we show that the set $$σ(S)\cup\{λ_i+λ_j:\;λ_i,λ_j\inσ(S),\,1\leqslant i<j\leqslant d\}$$ belongs to the point spectrum $σ_{\mathrm{pt}}(\mathcal{L})$ in $L^p$. We determine their eigenfunctions and show that they decay exponentially in space. As an application we analyze spinning soliton solutions which occur in the Ginzburg-Landau equation and compute their numerical spectra as well as associated eigenfunctions.

Wolf-Jürgen Beyn, Denny Otten

Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant evolution equa- tion. In numerical computations the freezing method takes advantage of this structure by splitting the evolution of the PDE into the dynamics on the underlying Lie group and on some reduced phase space. The approach raises a series of questions which were answered to a certain degree by the project: linear stability implies non-linear (asymp- totic) stability, persistence of stability under discretisation, analysis and computation of spectral structures, first versus second order evolution systems, well-posedness of partial differential algebraic equations, spatial decay of wave profiles and truncation to bounded domains, analytical and numerical treatment of wave interactions, relation to connecting orbits in dynamical systems. A further numerical problem related to this topic will be discussed, namely the solution of non-linear eigenvalue problems via a contour method.

Wolf-Jürgen Beyn, Elena Isaak, Raphael Kruse

This paper focuses on two variants of the Milstein scheme, namely the split-step backward Milstein method and a newly proposed projected Milstein scheme, applied to stochastic differential equations which satisfy a global monotonicity condition. In particular, our assumptions include equations with super-linearly growing drift and diffusion coefficient functions and we show that both schemes are mean-square convergent of order 1. Our analysis of the error of convergence with respect to the mean-square norm relies on the notion of stochastic C-stability and B-consistency, which was set up and applied to Euler-type schemes in [Beyn, Isaak, Kruse, J. Sci. Comp., 2015]. As a direct consequence we also obtain strong order 1 convergence results for the split-step backward Euler method and the projected Euler-Maruyama scheme in the case of stochastic differential equations with additive noise. Our theoretical results are illustrated in a series of numerical experiments.

Wolf-Jürgen Beyn, Elena Isaak, Raphael Kruse

This paper is concerned with the numerical approximation of stochastic ordinary differential equations, which satisfy a global monotonicity condition. This condition includes several equations with super-linearly growing drift and diffusion coefficient functions such as the stochastic Ginzburg-Landau equation and the 3/2-volatility model from mathematical finance. Our analysis of the mean-square error of convergence is based on a suitable generalization of the notions of C-stability and B-consistency known from deterministic numerical analysis for stiff ordinary differential equations. An important feature of our stability concept is that it does not rely on the availability of higher moment bounds of the numerical one-step scheme. While the convergence theorem is derived in a somewhat more abstract framework, this paper also contains two more concrete examples of stochastically C-stable numerical one-step schemes: the split-step backward Euler method from Higham et al. (2002) and a newly proposed explicit variant of the Euler-Maruyama scheme, the so called projected Euler-Maruyama method. For both methods the optimal rate of strong convergence is proven theoretically and verified in a series of numerical experiments.