dr. ir. F.W. Wubs
Department of Mathematics and Computing Science
9700 AK Groningen
The Netherlands
Tel: +31 (0)50 3633994
Fax: +31 (0)50 3633800
E-mail: wubs@math.rug.nl
See also http://www.rug.nl/staff/f.w.wubs
Dr. Wubs is associate professor in numerical
mathematics at the University of Groningen. He
studied applied mathematics at the
- Numerical Mathematics 1
- Numerical Mathematics 2
- Computational Methods of Science
- Finite Elements and Applications
In Short:
- Solution of linear systems with large sparse matrices (MRILU)
- Eigenvalue computations of large sparse matrices
- Discretization of PDEs
- Parallel and vector computing
- Fluid flow computations
- Bifurcation analysis
- Stability study of Ocean Flow Circulations
Description:
Solution
of large sparse linear systems In CFD problems often sparse linear systems
have to be solved, e.g. in explicit methods for incompressible flow a pressure
Poisson equation and in implicit methods the fully coupled Navier-Stokes
equations. The run time of codes for CFD problems is usually dominated by the
time needed for solving such a system. We were able to construct two efficient
methods for convection-diffusion equations: NGILU (Van der
Ploeg et al. 1996) and MRILU (Botta
&W. 1999). Both are incomplete LU factorizations of the system matrix with
special orderings for the unknowns and sophisticated dropping criteria, so the
product of the matrix factors L and U is not precisely the system matrix. NGILU was constructed for matrices that arise
from discretizations on regular cartesian grids and MRILU can handle matrices arising
from discretizations of unstructured grids since it
chooses self the ordering of the unknowns based on the magnitude of the entries
of the matrix. Like the almost classic
multi-grid methods, both show convergence independent of the number of
unknowns. So the amount of work to solve the system is linear in the number of
unknowns. In benchmark problems posed for a workshop at the
An interesting application of these methods are the ocean climate models as developed by IMAU. Since a decade we apply MRILU successfully to these problems (Wijer et al. 2003). Recently we worked out a special purpose method for the ocean climate model which speeded up the computations significantly; MRILU is used as a subsolver in this process.
Eigenvalue computations To study the stability of steady flows often a
(generalized) eigenvalue problem has to be solved.
One approach for this problem is the Jacobi-Davidson
method developed at Utrecht
University
Well-posed discretizations for PDEs A discretization of a PDE is called well-posed if the
solution of the
discretization is unique and varies continuously with
the coefficients in the PDE and associated boundary and initial conditions.
Moreover the discretization should be consistent to
the PDE. In many cases discretization of a well-posed
partial differential equation including boundary conditions is not a trivial
task. The discretization may be ill-posed leading to
instabilities during the numerical solution due to approximation errors and
round-off errors. This already starts
with second-order elliptic PDEs with suitable
boundary conditions. Though by itself these are well-posed and all eigenvalues of the associated operator together with the
homogeneous form of the boundary conditions are in the right half plane, the discretization may lead to a matrix which has eigenvalues left and right from zero or even may be
singular. The start of a stable discretization lies in a proper choice of the formulation of
the PDE. Usually one should start from a divergence form. From this form it is
easy to see whether the PDE, or the part of it where we are looking at, is
well-posed. Then the discretization should be
performed in such a way that this property is not lost, which leads to a stable
discretization. Such situations occurred in the ocean
model THCM (developed at
[Computational Mechanics and Numerical Mathematics]
- F.W. Wubs,
Numerical solution of the shallow-water equations, CWI Tract 49, Centre
for Mathematics and Computer Science,
- F.W. Wubs , Stabilization of explicit methods for hyperbolic partial differential equation, Int. j. numer. methods eng., 6, pp. 641-657, 1986.
- W.A. van der Veen and F.W. Wubs, A Hamiltonian approach to fairly low and fairly long gravity waves, Journal of Engineering Mathematics,29, pp. 329-345, 1995.
- A. van der Ploeg and E.F.F. Botta and F.W. Wubs, Grid-independent convergence based on preconditioning techniques, Multigrid methods IV, International series of numerical mathematics, pp. 333-344, 1994.
- E.F.F. Botta and F.W. Wubs, The convergence behaviour of iterative methods on severely stretched grids, Int. j. numer. methods eng., 36, pp. 3333-3350, 1993.
- E.F.F. Botta, A. van der Ploeg and F.W. Wubs, Nested grids ILU-decomposition (NGILU), J. Comput. Appl. Math., 66, pp. 515-526, 1996.
- E.F.F. Botta, K. Dekker, Y. Notay, A. van der Ploeg, C. Vuik, F.W. Wubs, P.M. de Zeeuw, How fast the Laplace equation was solved in 1995, App. Num. Math., 24, pp. 439-455, 1997.
- E.F.F. Botta and F.W. Wubs, Matrix Renumbering ILU: An effective algebraic multilevel ILU-preconditioner for sparse matrices, SIAM J. Matrix Anal. Appl., 20(4), pp 1007-1026, 1999.
- G.L.G. Sleijpen and F.W. Wubs, Effective preconditioning techniques for eigenvalue problems, August 1999. Revision november 2001
- F.W. Wubs, G. Tiesinga and A.E.P. Veldman, Bifurcation analysis of incompressible flow in a driven cavity, March 2000.
- A. Meijster and F.W. Wubs, Towards an Implementation of a Multilevel ILU preconditioner on Shared-Memory computers, LNCS 1823, pp 109-118, May 2000.
- H.A. Dijkstra, H. Oksuzoglu, F.W. Wubs and E.F.F. Botta, A fully implicit model of the three-dimensional thermohaline ocean circulation, JCP, 173, 685-715, 2001.
- G. Tiesinga, F.W. Wubs and A.E.P. Veldman, Bifurcation analysis of incompressible flow in a driven cavity by the Newton-Picard method, JCAM 140, 751-772, 2002.
- G.L.G. Sleijpen and F.W. Wubs, Exploiting Multilevel Preconditioning Techniques in Eigenvalue Computations, SISC 25(4), 1249-1272, 2003.
- W. Weijer, A.H. Dijkstra, H. Oksuzoglu, F.W. Wubs, and A.C. de Niet, A fully implicit model of the global ocean circulation, JCP, 192, 452-470, 2003
- F.W. Wubs, A.C. de Niet, and H.A. Dijkstra, The Performance of Implicit Ocean models on B- and C-grids, JCP, 211, 210-228, 2006.
- A.C. de Niet and F.W. Wubs, Two preconditioners for saddle point problems in fluid flows, Internat. J. Numer. Methods Fluids, 54, 355-377, 2007.
- A.C. de Niet and F.W. Wubs and H.A. Dijkstra and A. Terwisscha van Scheltinga, A tailored solver for bifurcations analysis of ocean-climate models, JCP 227(1), 654-679, doi:10.1016/j.jcp.2007.08.006
- A.C. de Niet and F.W. Wubs, Numerically stable LDLT-factorization of (F)-type saddle point matrices, RUG/UU, November 2006, submitted to IMA Journal of Numerical Analysis, report .
- E. Bernsen, H.A. Dijkstra, and F.W. Wubs, A method to reduce the spin-up time of ocean models, Ocean Modelling, accepted, 2007.
[Publications] of Computational Mechanics and Numerical Mathematics.
Comments via wubs@math.rug.nl
Last modified: August 7, 2008