Rudolf Wegmann

Cometary gas and plasma flow with detailed chemistry

Abstract This paper describes recent hydrodynamic and magnetohydrodynamic (MHD) simulations for the gas and plasma flow around a comet with detailed photo and chemical reaction network of 59 neutral and 76 ionized chemical species. The method allows for a separate energy balance of the electrons, separate flow of neutral gas, fast neutral atomic and molecular hydrogen, and plasma with momentum exchange by elastic collisions, mass-loading of the plasma flow by ion pick-up, and Lorentz-forces of the advected magnetic field. The contact surface or magnetopause is resolved. The simulation is applied to comet Halley at the time of the Giotto encounter. The results can be compared with the data from the Giotto mission as they become available. The chemical composition of the nucleus can be inferred from these data by iteration.

Cometary MHD and chemistry

Our magneto-hydrodynamical (MHD) and chemical comet-coma model has been applied to describe and analyze the plasma flow, the magnetic field and the ion abundances in comet P/Halley in a consistent manner. We assume the volatile composition to consist of 80% water and 20% carbon-, nitrogen-, oxygen-, and sulfur-compounds. The radius of the nucleus is 3.36 km. With hemispherical illumination, this is equivalent to the active area on the nucleus during the Giotto encounter. The physics and chemistry of the coma are modeled in great detail, including photoprocesses, gas-phase chemical kinetics, energy balance with a separate electron temperature, multifluid hydrodynamics with a transition to free molecular flow, fast-streaming atomic and molecular hydrogen, counter and cross-streaming of the ionized species relative to the neutral species in the coma-solar wind interaction region with momentum exchange by elastic collisions, mass-loading through ion pick-up, and Lorentz-forces of the advected magnetic field. A comparison of the results is made with the data from the Giotto mission, especially from the HIS ion mass spectrometer. We resolve the contact surface (magnetopause); its position is in agreement with observations. We also find the enhancement of the ion density just outside of the contact surface and agreement for the three groups in the ion mass spectra, particularly for the first group up to 21 amu.

Ein Iterationsverfahren zur konformen Abbildung

SummaryIn this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected regionG onto a regionH with smooth boundary. The method is based on a certain Riemann-Hilbert-problem. It turns out that this problem is the linearised version of a singular integral equation of the second kind. Hence the method is a Newton-method. Whenever the boundaries ofG andH are sufficiently smooth, convergence is locally quadratic.In caseG is the unit circle, the solution of the linearised problem can explicitly be represented in terms of integral-transformations. From this one derives a quadratically convergent Newton-like numerical method which avoids the numerical solution of systems of linear equations and therefore is in comparison with other methods, which are based on integral equations, rather economical in terms of computer time and storage requirements.

MHD-calculations for cometary plasmas

Abstract We present in this paper a relatively simple method for the numerical solution of the MHD-equations on a curvilinear grid. We consider the full 3D equations as well as a modified axisymmetric version. The difference scheme is based on the principle of forming backward differences along characteristics in the spatial variables. This method is used to calculate the interaction of the interplanetary magnetic field with the plasma flow around a comet. The MHD-equations are modified by source terms, which describe the transfer of mass, momentum and energy from a given background (the comet) to the plasma.

An iterative method for conformal mapping

Abstract In this paper we discuss an iterative method for the calculation of the boundary values of the conformal mapping of a simply connected region G onto a region H with smooth boundary. The method is based on a certain Riemann-Hilbert problem. It turns out that this problem is the linearized version of a singular integral equation of the second kind. Hence the method is a Newton method. Whenever the boundaries of G and H are sufficiently smooth, its convergence is locally quadratic. If G is the unit circle, the solution of the linearized problem can be represented explicitly in terms of integral transforms. From this one derives a quadratically-convergent Newton-like numerical method that avoids the numerical solution of systems of linear equations and therefore, in comparison with other methods based on integral equations, is quite economical in terms of computer time and storage requirements.

On nonnormal matrices

Abstract For the eigenvalues λi of an n × n matrix A the inequality ∑ i |λ i | 2 (‖A‖ 4 − 1 2 ‖D‖ 2 ) 1 2 is proved, where D ≔ AA ∗ − A ∗ A and ‖ ● ‖; denotes the euclidean norm. Conditions for equality are stated.

An iterative method for the conformal mapping of doubly connected regions

An iterative method is described for the determination of the conformal mapping of a circular annulus Rq ≔ {z:q < |z| < 1} onto a doubly connected region G with smooth boundary curves. The method is analogous to the method of [7] for simply connected regions. It requires in each step the solution of a Riemann-Hilbert problem on Rq. This problem can be solved explicitly in terms of a generalized conjugation operator K. A degeneracy of this problem leads in a natural way to an adjustment of the parameter q in each step. If the boundary curves of G admit parametrizations by functions ην with second derivatives which are Holder continuous with exponent μ, 0 < μ ⩽ 1, then the method converges locally in the Sobolev space W of 2π-periodic absolutely continuous functions with square-integrable derivatives. The order of convergence is at least 1 + μ. The convergence is quadratic if the ην have Lipschitz-continuous second derivatives. For the numerical implementation of the method the operator K can be approximated by a Wittich type method, which can be performed very effectively using FFT. A calculation with N = 2m grid points on the computer requires storage of the order O(N) and computing time O(mN). The results of some test calculations are reported and compared with results of calculations with the method of Theodorsen-Garrick.

The asymptotic eigenvalue-distribution for a certain class of random matrices

Abstract For an n × n Hermitean matrix A with eigenvalues λ1, …, λn the eigenvalue-distribution is defined by G(x, A) := 1 n · number {λi: λi ⩽ x} for all real x. Let An for n = 1, 2, … be an n × n matrix, whose entries aik are for i, k = 1, …, n independent complex random variables on a probability space (Ω, R , p) with the same distribution Fa. Suppose that all moments E | a | k, k = 1, 2, … are finite, E a=0 and E | a | 2. Let M(A)= ∑ σ=1 s θ σ P σ (A,A ∗ ) with complex numbers θσ and finite products Pσ of factors A and A ∗ (= Hermitean conjugate) be a function which assigns to each matrix A an Hermitean matrix M(A). The following limit theorem is proved: There exists a distribution function G0(x) = G1x) + G2(x), where G1 is a step function and G2 is absolutely continuous, such that with probability 1 G(x, M( A n n 1 2 )) converges to G0(x) as n → ∞ for all continuity points x of G0. The density g of G2 vanishes outside a finite interval. There are only finitely many jumps of G1. Both, G1 and G2, can explicitly be expressed by means of a certain algebraic function f, which is determined by equations, which can easily be derived from the special form of M(A). This result is analogous to Wigners semicircle theorem for symmetric random matrices ( E. P. Wigner, Random matrices in physics, SIAM Review 9 (1967) , 1–23). The examples A r A ∗r , A r + A ∗r , A r A ∗r ± A ∗r A r , r = 1, 2, …, are discussed in more detail. Some inequalities for random matrices are derived. It turns out that with probability 1 the sharpened form lim sup n→∞ ∑ i=1 n |γ i (n) | 2 ⧹‖A n ‖ 2 ⩽ 0.8228… of Schurs inequality for the eigenvalues λi(n) of An holds. Consequently random matrices do not tend to be normal matrices for large n.

Discretized versions of Newton type iterative methods for conformal mapping

Abstract It is shown that for nearly circular regions discretized versions of the Newton type methods of Wegmann (1978) and Hubner (1986) converge locally to fixed points. Convergence is linear. The rates can be determined approximately for several standard regions. The dominant operator acts only on a subspace of high-order harmonics. Therefore under conditions of smoothness and/or symmetry convergence can be much faster. The fixed points satisfy a discrete version of Lavrentevs variational principle. Therefore the resulting approximations for conformal mapping are essentially as accurate as Wittichs approximation for the conjugation operation.

Discrete Riemann-Hilbert problems, interpolation of simply closed curves, and numerical conformal mapping

Abstract Let G be a simply connected region with 0 ϵ G and with a twice Lipschitz continuously differentiable boundary curve, Γ, and let z μ , μ = 1,…, N , be an even number of N = 2 n equidistant grid points on the unit circle {sfnc z sfnc = 1} with z 1 = 1. Then there exists for all sufficiently large N a polynomial P n of degree n + 1, normalized by the condition that the coefficient p 0 = 0, and the coefficients p 1 and p n +1 are real, such that P n satisfies the interpolation condition P n ( z μ ) ϵ Γ for all μ = 1,…, N . In a neighbourhood of the normalized conformal mapping function Φ there is exactly one such interpolating polynomial. The sequence of these P n converges to the conformal mapping function Φ as n → ∞. If Γ is three times Lipschitz continuously differentiable, then the P n are also conformal mappings of the unit circle onto regions, which approximate G . An important tool for the theoretical investigation is a discrete analogon of the Riemann—Hilbert problem. We present two fast procedures for the numerical solution of the discrete Riemann—Hilbert problem: A conjugate gradient method with computational cost O( N log N ) and a Toeplitz matrix method with cost O( N log 2 N ). Using this, one can calculate the interpolating polynomials by a Newton method and in this way obtains very effective methods for the numerical approximation of the conformal mapping function.