Talks

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).

In CFD, Atmospheric Sciences and Computational Aeroacoustics, many problems involve regions of discontinuity. When used to solve problems involving regions of shocks, dispersive schemes give rise to oscillations while dissipative schemes cause smearing, close to these regions of sharp gradients.

Based on the results of the 1-D shallow water problem, when solved using MCLF2, we observe that different cfl numbers yield results with different amount of dispersion and dissipation. This led us to devise a technique in order to locate the cfl number at which we can obtain results with efficient shock-capturing properties. This new technique involves the control of numerical effects of dispersion and dissipation in numerical schemes. We baptise this technique as Curbing of Dispersion by Dissipation for Efficient Shock-capturing, CDDES. The cfl number at which dissipation curbs dispersion optimally is then located. It is termed as the optimal cfl.

We extend the concept of CDDES to that of Minimised Integrated Square Difference Error,(MISDE). The latter is an improved technique over the CDDES technique since it can be used to obtain two optimal parameters which are generally the cfl number and another variable, for efficient-shock capturing. Another technique of optimisation is devised which enables better control over the grade and balance of oscillation and dissipation to optimise parameters which regulate dispersion and dissipation effects. This technique is baptised as Minimised Integrated Exponential Error for Low Dispersion and Low Dissipation, (MIEELDLD) and has advantages over the previous technique, MISDE.

A dimension reduction method called Discrete Empirical Interpolation (DEIM) will be presented and shown to dramatically reduce the computational complexity of the popular Proper Orthogonal Decomposition (POD) method for constructing reduced-order models for unsteady and/or parametrized nonlinear partial differential equations (PDEs). In the presence of a general nonlinearity, the standard POD-Galerkin technique reduces dimension in the sense that far fewer variables are present, but the complexity of evaluating the nonlinear term remains that of the original problem.

I will describe DEIM as a modification of POD that reduces the complexity as well as the dimension of general nonlinear systems of ordinary differential equations (ODEs). It is, in particular, applicable to ODEs arising from finite difference discretization of unsteady time dependent PDE and/or parametrically dependent steady state problems. Our contribution is a greatly simplified description of Empirical Interpolation in a finite dimensional setting. The method possesses an error bound on the quality of approximation. An application of DEIM to a finite difference discretization of the 1-D FitzHugh-Nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long-time integration that fully captures non-linear limit cycle behavior. We also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results.

The lecture will be concerned with the simulation of inviscid and viscous compressible flow in time dependent domains. The motion of the boundary of the domain occupied by the fluid is taken into account with the aid of the ALE (Arbitrary Lagrangian-Eulerian) formulation of the Euler and Navier-Stokes equations describing compressible flow. The system of the governing equations is discretized in space by the discontinous Galerkin method. The time discretization is based on a semi-implicit linearized time stepping scheme, which leads to the solution of a linear algebraic system on each time level. As a result we get an efficient and robust numerical process. The applicability of the developed method will be demonstrated by some computational results obtained for flow in a channel with a moving wall and past an oscillating airfoil.

These results were obtained in cooperation with Vaclav Kucera and Jaroslava Prokopova from Charles University in Prague, Faculty of Mathematics and Physics, and Jaromir Horacek from Institute of Thermomechanics of Academy of Sciences of the Czech Republic.

The Induced Dimension Reduction (IDR) method is a Krylov space method for solving linear systems that was first developed by Sonneveld around 1979 and documented on three and a half pages of a 1980 proceedings paper by Wesseling and Sonneveld. Soon after IDR, Sonneveld introduced his widely applied Conjugate Gradient Squared (CGS) algorithm. Then, in 1990, van der Vorst suggested Bi-CGSTAB that he claimed to improve both those methods.

Bi-CGSTAB has become a method of choice for nonsymmetric linear systems, and it has been generalized in various ways in the hope of further improving its reliability and speed. Among these generalizations there is the ML(k)BiCGSTAB method of Yeung and Chan, which in the framework of block Lanczos methods can be understood as a variation of Bi-CGSTAB with right-hand side block size 1 and left-hand side block size k.

In 2007 Sonneveld and van Gijzen reconsidered IDR and generalized it to IDR(s), claiming that IDR is equally fast but preferable to Bi-CGSTAB, and that IDR(s) may be much faster than IDR = IDR(1). It turned out that IDR(s) is closely related to BiCGSTAB if s = 1 and to ML(s)BiCGSTAB if s > 1. In 2008, a new, particularly ingenious and elegant variant of IDR(s) has been proposed by the same authors.

In this talk we first try to explain the basic, seemingly quite general IDR approach, which differs completely from traditional approaches to Krylov space methods. Then we compare the basic properties of the above mentioned methods and discuss some of their connections.