# Reports of Scientific Computing

Research group: Scientific Computing (MAS2.1)

*Research group: Scientific Computing (MAS2.1)*

*Using domain decomposition in the Jacobi-Davidson method*

M. Genseberger, G.L.G. Sleijpen, H.A. van der Vorst

2000, MAS-R0029, ISSN 1386-3703

The Jacobi-Davidson method is suitable for computing solutions of large*n*-dimensional eigenvalue problems. It needs (approximate) solutions of specific*n*-dimensional linear systems. Here we propose a strategy based on a nonoverlapping domain decomposition technique in order to reduce the wall clock time and local memory requirements.

Download the pdf-file*Numerical solution of steady free-surface Navier-Stokes flow*

E.H. van Brummelen

2000, MAS-R0018, ISSN 1386-3703

Numerical solution of flows that are partially bounded by a freely moving boundary is of great practical importance, e.g., in ship hydrodynamics. The usual time integration approach for solving steady viscous free surface flow problems has several drawbacks. Instead, we propose an efficient iterative method, which relies on a different but equivalent formulation of the free surface flow problem, involving a so-called quasi free-surface condition.

Download the pdf-file*Computing probabilistic bounds for extreme eigenvalues of symmetric matrices with the Lanczos method*

J.L.M. van Dorsselaer, M.E. Hochstenbach, H.A. van der Vorst

1999, MAS-R9934, ISSN 1386-3703

In many applications it is important to have reliable approximations for the extreme eigenvalues of a symmetric or Hermitian matrix. A method which is often used to compute these eigenvalues is the Lanczos method. Unfortunately it is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues -- even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we derive probabilistic bounds for the extreme eigenvalues.

Download the pdf-file*On the representation of functions and finite difference operators on adaptive sparse grids*

P.W. Hemker, F. Sprengel

1999, MAS-R9933, ISSN 1386-3703

In this paper we describe methods to approximate functions and differential operators on adaptive sparse grids. We distinguish between several representations of a function on the sparse grid, and we describe how finite difference (FD) operators can be applied to these representations.

Download the pdf-file*Application of the over-set grid technique to a model singular perturbation problem*E.D. Havik, P.W. Hemker, W. Hoffmann

1999, MAS-R9931, ISSN 1386-3703

The numerical solution of a singularly perturbed problem, in the form of a two-dimensional convection-diffusion equation, is studied by using the technique of over-set grids. For this purpose the Overture software library is used. The selection of component grids is made on basis of asymptotic analysis. The behavior of the solution is studied for a range of small diffusion parameters. Also the possibilities of rotating the grid with the convection direction is considered.

Download the pdf-file*The sparse-grid combination technique applied to time-dependent advection problems*B. Lastdrager, B.Koren, J.G. Verwer

1999, MAS-R9930, ISSN 1386-3703

In the numerical technique considered in this paper, time-stepping is performed on a set of semi-coarsened space grids. At given time levels the solutions on the different space grids are combined to obtain the asymptotic convergence of a single, fine uniform grid. We present error estimates for the two-dimensional spatially constant-coefficient model problem and discuss numerical examples.

Download the pdf-file*Some remarks on multilevel algorithms for finite difference discretizations on sparse grids*

F. Sprengel

1999, MAS-R9924, ISSN 1386-3703

In this paper, we propose some algorithms to solve the system of linear equations arising from the finite difference discretization on sparse grids. For this, we will use the multilevel structure of the sparse grid space or its full grid subspaces, respectively.

Download the pdf-file*Analysis of the incompressible Navier-Stokes equations with a quasi free-surface condition*E.H. van Brummelen

1999, MAS-R9922, ISSN 1386-3703

Numerical solution of free-surface flows with a free-surface that can be represented by a height-function, is of great practical importance. Dedicated methods have been developed for the efficient solution of steady free-surface potential flow. These methods solve a sequence of sub-problems, corresponding to the flow equations subject to a quasi free-surface condition. For steady free-surface Navier-Stokes flow, such dedicated methods do not exist. In the present report we propose an extension to Navier-Stokes flow of an iterative method which has been applied successfully to steady free-surface potential flow.

Download the pdf-file*Adaptive sparse-grid combination-solutions for a singular perturbation problem*

J. Noordmans

1999, MAS-R9916, ISSN 1386-3703

In this paper we show how, under minimal conditions, a combination extrapolation can be introduced for an adaptive sparse grid. We apply this technique for the solution of a two-dimensional model singular perturbation problem, defined on the domain exterior of a circle.

Download the pdf-file (10MB)*A Godunov-type scheme with applications in hydrodynamics*

E.H. van Brummelen

1998, MAS-R9829, ISSN 1386-3703

In spite of the absence of shock waves in most hydrodynamic applications, sufficient reason remains to employ Godunov-type schemes in this field. In the instance of two-phase flow, the shock capturing ability of these schemes may serve to maintain robustness and accuracy at the interface. Moreover, approximate Riemann solvers have greatly relieved the initial drawback of computational expensiveness of Godunov-type schemes. In the present work we develop an Osher-type flux-difference splitting approximate Riemann solver and we examine its application in hydrodynamics. Actual computations are left to future research.

Download the pdf-file*Some error estimates for periodic interpolation of functions from Besov spaces*W. Sickel, F. Sprengel

1998, MAS-R9826, ISSN 1386-3703

Using periodic Strang--Fix conditions, we can give an approach to error estimates for periodic interpolation on equidistant and sparse grids for functions from certain Besov spaces.

Download the pdf-file*Interpolation on sparse Gauss--Chebyshev grids in higher dimensions*

F. Sprengel

1998, MAS-R9824, ISSN 1386-3703

In this paper, we give a unified approach to error estimates for interpolation on sparse Gauß--Chebyshev grids for multivariate functions from Besov--type spaces with dominating mixed smoothness properties. The error bounds obtained for this method are almost optimal for the considered scale of function spaces.

Download the pdf-file*Error analysis for function representation by the sparse-grid combination technique*B. Lastdrager, B. Koren

1998, MAS-R9823, ISSN 1386-3703

Detailed error analyses are given for sparse-grid function representations through the combination technique. Two- and three-dimensional, and smooth and discontinuous functions are considered, as well as piecewise-constant and piecewise-linear interpolation techniques. Where appropriate, the results of the analyses are verified in numerical experiments. Instead of the common vertex-based function representation, cell-centered function representation is considered. Explicit, pointwise error expressions for the representation error are given, rather than order estimates. The paper contributes to the theory of sparse-grid techniques.

Download the pdf-file*Alternative correction equations in the Jacobi-Davidson method*

M. Genseberger, G.L.G. Sleijpen

1998, MAS-R9816, ISSN 1386-3703

The correction equation in the Jacobi-Davidson method is effective in a subspace orthogonal to the current eigenvector approximation, while for the continuation of the process only vectors orthogonal to the search subspace are of importance. Such a vector is obtained by orthogonalizing the (approximate) solution of the correction equation against the search subspace. As an alternative, a variant of the correction equation can be formulated that is restricted to the subspace orthogonal to the current search subspace. In this paper, we discuss the effectivity of this variant.

Download the pdf-file*Note on PARNASSOS, a Navier-Stokes method for ship-stern flows*

B. Koren

1998, MAS-R9808, ISSN 1386-3703

In this report, analyses are made of some of the numerical techniques implemented in MARIN's viscous ship-hydrodynamics software PARNASSOS. Suggestions are given to improve the robustness of PARNASSOS and -- where appropriate -- its accuracy and computational efficiency. The effects to be expected from some of the proposed modifications are analyzed as well.

Download the pdf-file*Restarting parallel Jacobi-Davidson with both standard and harmonic Ritz values*M.Nool, A. van der Ploeg

1998, MAS-R9807, ISSN 1386-3703

We study the Jacobi-Davidson method for the solution of large generalized eigenproblems as they arise in MagnetoHydroDynamics. We have combined Jacobi-Davidson (using standard Ritz values) with a shift and invert technique. We apply a complete LU decomposition in which reordering strategies based on a combination of block cyclic reduction and domain decomposition result in a well-parallelizable algorithm. Moreover, we describe a variant of Jacobi-Davidson in which harmonic Ritz values are used. In this variant the same parallel LU decomposition is used, but this time as a preconditioner to solve the `correction` equation.

Download the pdf-file*A parallel Jacobi-Davidson method for solving generalized eigenvalue problems in linear magnetohydrodynamics*M. Nool, A. van der Ploeg

1997, MAS-R9733, ISSN 1386-3703

We study the solution of generalized eigenproblems generated by a model which is used for stability investigation of tokamak plasmas. The eigenvalue problems are of the form*A x = λ B x*, in which the complex matrices*A*and*B*are block tridiagonal, and*B*is Hermitian positive definite. The Jacobi-Davidson method appears to be an excellent method for parallel computation of a few selected eigenvalues, because the basic ingredients are matrix-vector products, vector updates and inner products. The method is based on solving projected eigenproblems of order typically less than 30.

Download the pdf-file*A level-set method for moving material-void interfaces*B. Koren, A.C.J. Venis

1997, MAS-R9731, ISSN 1386-3703

This report is a feasibility study of a level-set method for the computation of moving interfaces, in an Eulerian formulation. The report briefly introduces level-set methods and focuses on the development of such a method for moving material-void interfaces. Results are presented for illustrative model problems. As concerns its ability to improve the geometrical resolution of free boundaries, the level-set method appears to perform excellently. Concerning the improvement of other than merely geometrical free-boundary properties, the method performs very well for downstream-facing fronts and is promising for upstream-facing ones.

Download the pdf-file*Convergence results for 3D sparse grid approaches*J. Noordmans, P.W. Hemker

1997, MAS-R9713, ISSN 1386-3703

The convergence behaviour is investigated of solution algorithms for the anisotropic Poisson problem on partially ordered, sparse families of regular grids in 3D. In order to study multilevel techniques on sparse families of grids, first we consider the convergence of a two-level algorithm that applies semi-coarsening successively in each of the coordinate directions. This algorithm shows good convergence, but recursive application of the successive semi-coarsening is not sufficiently efficient.

Download the pdf-file- Numerieke berekeningen aan een luchtstroming in een spleetblazer, uit het oogpunt van trekkrachtgeneratie in een nonwoven-produktieproces

D. Lanser

1997, MAS-N9702, ISSN 1386-3703

Het onderzoek is erop gericht geweest door numerieke berekeningen aan de luchtstroming in verschillende blazergeometrieën meer inzicht te verkrijgen in de trekkrachtverhogende werking van de verbreding. De luchtstroming is daartoe gemodelleerd met behulp van de 2D Navier-Stokes vergelijkingen. Aangenomen is dat de perslucht een ideaal gas is.

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