Rita Meyer-Spasche

Computations of the axisymmetric flow between rotating cylinders

We study Taylor vortex flows by solving the steady axisymmetric Navier-Stokes. equations in the primitive variables (u, v, w, p). Fourier expansions in z, the axial direction, and centered finite differences in r, the radial direction, are used. The resulting discretized equations are solved using the pseudoarclength continuation methods of Keller (in “Applications of Bifurcation Theory” (P. Rabinowitz, Ed.), pp. 359–384, Academic Press, New York, 1977.), which are designed to detect bifurcations. In this way we accurately determine the first branch of Taylor vortex solutions bifurcating from Couette flow for both a wide and a narrow gap. Agreement with experiments is extremely good for the wide gap case and solutions are obtained for a larger range of Reynolds numbers than previously reported.

Difference schemes of optimum degree of implicitness for a family of simple ODEs with blow-up solutions

Abstract It has been speculated that the linearly implicit linearized trapezoidal rule should be especially useful for computing blow-up solutions since it features discrete blow-up. We study the question of how implicit a scheme should be from several different points of view: we discuss further the properties of the linearized trapezoidal and of its discrete blow-up, we give exact schemes for a family of ODEs with polynomial nonlinearity, and that shows ‘the optimum degree of implicitness’ they need; we show that other standard schemes (the trapezoidal rule itself and the implicit midpoint rule) are exact on certain differential equations; we compare for several schemes and different nonlinearities the size of the leading error terms; and we briefly discuss two ways of adapting the degree of implicitness of a scheme to a given differential equation.

Analysis of electron trajectories in a gyrotron resonator

A detailed mathematical analysis of the equation describing the electron interaction with the high-frequency field in a gyrotron resonator is presented. Electron trajectories in the phase space are classified. It is proven that in the cold-cavity approximation when the high-frequency field is represented by a Gaussian-type function, the solutions of the gyrotron equation are asymptotically equal to the solutions of the corresponding unforced equation. This means that chaos, which, in principle, can develop in a resonator for some values of control parameters, can be only transient, i.e., electrons again follow regular trajectories once they leave the interaction space. Understanding of the distribution of electron trajectories is important, both from the theoretical and the practical point of view. As an example, detailed numerical computations of electron trajectories are performed for those values of control parameters which correspond to the maximum efficiency.

On the Decay of Symmetric Toroidal Dynamo Fields

The “anti-dynamo” theorems for toroidal magnetic fields with axisymmetry and plane symmetry are generalized to the case of a compressible, time-dependent flow in a fluid with arbitrary conductivity

Computation of transitions in Taylor vortex flows

The very reliable numerical methods of [15, 18] were used for the investigation of axisymmetric flows between infinitely long rotating cylinders. Solutions consisting of vortices of different sizes were detected. Varying both the wavelength of the vortices and the Reynolds number, we showed that the solutions are intermediate states between solutions with equally sized vortices (transitions from 2 to 4 and from 2 to 6 vortices).ZusammenfassungDie sehr zuverlässigen numerischen Methoden von [15, 18] wurden zur Berechnung von axialsymmetrischen Strömungen zwischen unbeschränkt langen, rotierenden Zylindern benutzt. So wurden Lösungen gefunden, die aus Wirbeln unterschiedlicher Größen bestehen. Durch Variation der Wellenlänge der Wirbel und der Reynoldszahl wurde gezeigt, daß diese Lösungen Übergangszustände sind zwischen verschiedenen Lösungen mit Wirbeln einheitlicher Größe (Übergänge von 2 zu 4 und von 2 zu 6 Wirbeln).

A finite difference procedure for a class of free boundary problems

Finite difference schemes loose accuracy when free boundaries cross over rectangular grids. For a class of second-order equations, the leading error term at such a boundary can be eliminated by a simple correction strategy. This procedure works in any number of space dimensions and offers an alternative to (more costly and complicated) adaptive grid techniques.

Numerical Treatment of Dirichlet Problems with Several Solutions

MHD equilibrium calculations involve elliptic boundary value problems (1) Lu = F(u), Bu = 0 having several solutions. It is explained here why all solutions should be calculated, i.e. including those which are unstable in the sense of Liapunov if they are regarded as equilibrium solutions of (2) \({u_t} = - Lu{\mkern 1mu} + {\mkern 1mu} F(u),{\mkern 1mu} \tilde B(u){\mkern 1mu} = {\mkern 1mu} 0.\) These “unstable” solutions cannot be calculated by certain iterative methods, in principle. The objective here is to find numerical methods allowing efficient computation of the “unstable” solutions. Convergence theorems for such methods will be reported elsewhere by the author. This paper presents solutions and bifurcation diagrams computed by them.

A method of solving the stability problem for complex matrices

AbstractThe aim of this paper is to determine the location of eigenvalues of a general complex matrix without explicitly computing them. Essentially, the method consists in finding a Hermitian matrixH which has the same inertia as the given matrixM, and determining the inertia ofH. H is found as the solution of the Liapunov equation

Discretization errors at free boundaries of the Grad-Schlüter-Shafranov equation

H\tilde M + \tilde M^* H = 2I