Talks

Bekanntlich liefert ein Schriitt $(u,\theta) \mapsto u_+^{InvIt}$ der Inversen Iteration für das nichtlineare Eigenwertproblem $T(\lambda)x=0$ dieselbe Richtung wie ein Schritt $(u,\theta) \mapsto (u_+^{Newt},\theta_+^{Newt})$ des Newtonverfahrens für das erweiterte System $T(\lambda)x=0,\;w^Hx=1$ mit einem geeigneten Skalierungsvektor $w$, d.h., es gilt $\mbox{span}\,\{u_+^{InvIt}\}=\mbox{span}\,\{u_+^{Newt}\}$. Es liegt daher nahe, zur Abschätzung der Verbesserung der Eigenvektorapproximation $u$ durch die Inverse Iteration Newton-Techniken zu verwenden. Es wird gezeigt, dass dies zu genauen Abschätzungen führt, wenn explizit mit dem Restglied zweiter Ordnung gearbeitet und dessen spezielle Produktstruktur berücksichtigt wird wie das von \textsc{Heinz Unger} [50] erstmalig (und ohne publizierten Beweis) für das lineare Problem $T(\lambda)=A-\lambda I$ getan worden ist.

Durch Kombination mit neuen Abschätzunegn für das nichtlineare klassische bzw. verallgemeinerte Rayleigh-Funktional läßt sich dann einfach die quadratische Konvergenz
der nichtlinearen Rayleigh-Funktional-Iteration wie auch die kubische Konvergenz der nichtlinearen Verallgemeinerung der zweiseitigen Ostrowskischen Rayleigh-Quotienten-Iteration herleiten.

The motion of one or several rigid bodies in a viscous fluid occupying a bounded domain ­$\Omega in R^3$ represents an interesting theoretical problem featuring, among others, possible contacts of two or more solid objects. We consider the motion of several rigid bodies in a non-Newtonian fluid of a power-law type. Our main result establishes the existence of global-in-time solutions of the associated evolutionary system, when collisions of two or more rigid objects do not appear in a finite time unless they were present initially.

We solve a particular system of nonlinear ODEs defined on the two different components of the real line connected by the nonlinear contact condition
\[
w^\prime =h^\prime \;,\qquad h=\psi(w)\qquad\text{at the point $\,x=0\,$}.
\]
We show that, for a prescribed power-law nonlinearity $\psi$ and using the solution $(w,h)$, a self-similar solution to the porous medium equation in the two-component domain can be constructed.

The detection of a Hopf bifurcation in a large scale dynamical system that depends on a physical parameter often consists of computing the right-most eigenvalues of a sequence of large sparse eigenvalue problems. Guckenheimer et. al. (SINUM, 34, (1997) pp. 1-21) proposed a method that computes a value of the parameter that corresponds to a Hopf point without actually computing right-most eigenvalues. This method utilises a certain sum of Kronecker products and involves the solution of matrices of squared dimension, which is impractical for large scale applications. However, if good starting guesses are available for the parameter and the purely imaginary eigenvalue at the Hopf point, then efficient algorithms are available. In this paper, we propose a method for obtaining such good starting guesses, based on finding purely imaginary eigenvalues of a two-parameter eigenvalue problem (possibly arising after a linearisation process). The problem is formulated as an inexact inverse iteration method that requires the solution of a sequence of Lyapunov equations with low rank right hand sides. It is this last fact that makes the method feasible for large systems. The power of our method is tested on numerical examples.

The conventional SOR (Successive Over-Relaxation) method originated from the dissertation by D. Young in 1950. After that, the SOR method has been often used for the solution of problems which stem from various applications. The SOR method, however, has many issues on possibility of the solution because of no robustness of convergence of the SOR method.

Recently Sonneveld and van Gijzen brought epoch-making and renewed interest in the Induced Dimension Reduction (IDR) method in 2008. In addition, the Bi_IDR(s) method which was proposed by them is more elegant and stable than IDR(s) method. Furthermore, in 2009, IDR(s)Stab(L) and GBiCGStab(s,L) methods were independently proposed as one of the generalized version of IDR(s) method with polynomial of high degree L by Sleijpen and Tanio et al.

In my talk, we extend IDR Theorem to designing of the residual of the Jacobi, Gauss-Seidel and SOR methods, and accelerate their convergence rate and robustness. Through numerical experiments, we make clear improvement of performance of IDR-based Jacobi, Gauss-Seidel and SOR methods with parameters.

In this talk I will briefly review the generalized Riemann problem (GRP) method for compressible fluid flows. There were originally two versions of this method:
Lagrangian and Eulerian. The latter is always derived via a passage from the former. In our recent efforts, we developed a direct Eulerian GRP method using the ingredient of Riemann invariants. The main advantage is (1) to avoid the passage from the Lagrangian to Eulerian and thus easily to be extended into multidimensional cases; (2) treat sonic cases easily; and (3) conveniently combine with other techniques such as adaptive meshes.
We will also report some stability, convergence properties, and applications to shallow water equations on the sphere (earth).