Jens Lorenz

Stable attracting sets in dynamical systems and in their one-step discretizations

We consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set Λ of arbitrary shape. Under the assumption of uniform asymptotic stability of Λ in the sense of Lyapunov, we show that discretized versions of the dynamical system involving one-step numerical methods have nearby attracting sets Λ(h), which are also uniformly asymptotically stable. Our proof uses the properties of a Lyapunov function which characterizes the stability of Λ.

Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations

We study the slightly compressible Navier-Stokes equations. We first consider the Cauchy problem, periodic in space. Under appropriate assumptions on the initial data, the solution of the compressible equations consists-to first order-of a solution of the incompressible equations plus a function which is highly oscillatory in time. We show that the highly oscillatory part (the sound waves) can be described by wave equations, at least locally in time. We also show that the bounded derivative principle is valid; i.e., the highly oscillatory part can be suppressed by initialization. Besides the Cauchy problem, we also consider an initial-boundary value problem. At the inflow boundary, the viscous term in the Navier-Stokes equations is important. We consider the case where the compressible pressure is prescribed at inflow. In general, one obtains a boundary layer in the pressure; in the velocities a boundary layer is not present to first approximation.

On the scaling of multidimensional matrices

Abstract Elementary proofs are given for theorems of Bapat and Raghavan on the scaling of nonnegative multidimensional matrices. Theorems of Sinkhorn and of Brualdi, Parter, and Schneider are derived as corollaries. For positive two-dimensional matrices, Hilberts projective metric and a theorem of G. Birkhoff are used to prove that Sinkhorns original iterative procedure converges geometrically; the ratio of convergence is estimated from the given data.

Numerical calculation of invariant tori

The problem of computing a smooth invariant manifold for a finite-dimensional dynamical system is considered. In this paper, it is assumed that the manifold can be parameterized over a torus in terms of a subset of the system variables. The approach used here then involves solving a system of partial differential equations subject to periodic boundary conditions.The resulting numerical approach is analyzed and contrasted with some previously tested ones, and several methods of implementation are considered. Numerical results are given for the forced van der Pol equation and for a system of two linearly coupled oscillators.

Stability of traveling waves: dichotomies and eigenvalue conditions on finite intervals

If a traveling wave is stable or unstable depends essentially on the spectrum of a differential operator P obtained by linearization. We investigate how spectral properties of the all-line operator P are related to eigenvalues of finite-interval . Here R is a. linear boundary operator, for which we will derive determinant conditions, and the x-interval is assumed to be sufficiently large. Under suitable assumptions, we show (a) resolvent estimates for large s; (b) if s is in the resolvent of the all-line operator P, then s is also in the resolvent for finite-interval BVPs; (c) eigenvalues of P lead to approximating eigenvalues on finite intervals. These results allow to study the stability question for traveling waves by investigating eigenvalues of finite-interval problems. We give applications to the FitzHugh-Nagumo system with small diffusion and to the complex Ginzburg-Landau equations.

Computation of invariant tori by the method of characteristics

In this paper we present a technique for the numerical approximation of a branch of invariant tori of finite-dimensional ordinary differential equations systems. Our approach is a discrete version of the graph transform technique used in analytical work by Fenichel [Indiana Univ. Math. J., 21 (1971), pp. 193–226]. In contrast to our previous work [L. Dieci, J. Lorenz, and R. D. Russell, SIAM J. Sci. Statist. Comput., 12 (1991), pp. 607–647], the method presented here does not require a priori knowledge of a suitable coordinate system for the branch of invariant tori, but determines and updates such a coordinate system during a continuation process. We give general convergence results for the method and present its algorithmic description. We also show how the method performs on two physically important nonlinear problems, a system of two coupled oscillators and the forced van der Pol oscillator. In the latter case, we discuss some modifications needed to approximate an invariant curve for the Poincare map.

An analysis of the petrov—galerkin finite element method

Abstract Petrov-Galerkin finite element methods, using different test and trial functions, are applied to the solution of an unsymmetric two-point boundary value problem intended to simulate certain aspects of convection-diffusion problems. For a specified space of trial functions we utilise an energy error bound to optimize this class of methods over a family of test spaces. The optimized method performs well provided the asymmetry in the differential operator does not lead to boundary layers in the solution. Following an analysis of the boundary layer behaviour of the continuous problem, L -splines are introduced, and, by studying their behaviour for coarse meshes, we are able to modify the original schemes to produce so-called “disconnected” finite element methods. Even for coarse meshes, when no nodes occur in the boundary layer, the accuracy at all nodal points is good. This would make them good candidates for application in more general situations.

Center manifolds of dynamical systems under discretization

We consider an autonomous dynamical system discretized by a one-step method. The point z = 0 is assumed to be fixed under the continuous and the discrete flows. We allow z = 0 to be non-hyperbolic. The continuous system has a center-unstable manifold and we show the existence of approximating invariant manifolds for the discretizations. The manifolds for the continuous and the discrete systems share the property of being locally attracting at an exponential rate; the dynamics inside the manifolds can differ qualitatively, however, for all step-sizes h.

Zur Inversmonotonie diskreter Probleme

SummaryThis paper describes sufficient conditions for a real square matrixA=(aij) to have a nonnegative inverse. It is not assumed thataij≦0 fori≠j. We indicate several applications to matricesA that occur in finite-difference and finiteelement methods for boundary-value problems.

Hermite methods for hyperbolic initial-boundary value problems

We study arbitrary-order Hermite difference methods for the numerical solution of initial-boundary value problems for symmetric hyperbolic systems. These differ from standard difference methods in that derivative data (or equivalently local polynomial expansions) are carried at each grid point. Time-stepping is achieved using staggered grids and Taylor series. We prove that methods using derivatives of order m in each coordinate direction are stable under m-independent CFL constraints and converge at order 2m + 1. The stability proof relies on the fact that the Hermite interpolation process generally decreases a seminorm of the solution. We present numerical experiments demonstrating the resolution of the methods for large m as well as illustrating the basic theoretical results.