New mathematical techniques for fluid flow simulation

1
New mathematical techniques for fluid flow
simulation David W. Cook II Department of
Mathematics, College of Arts and
Sciences Tennessee Technological University,
Cookeville, Tennessee Advisor Dr. Sabine Le Borne

• Introduction
• Computational fluid dynamics (CFD) is the
  simulation on a computer of how fluids flow in an
  environmenthow water flows over a waterfall, for
  example.

• Results
• The major accomplishment of this project has been
  the development and implementation of the H-QR
  factorization technique to construct a
  well-conditioned, approximate null basis Z.
• The numerical tests have been performed on a Dell
  690n workstation (2.33 GHz, 32 GB memory) using
  the H-matrix library HLIB (available at
  http//www.hlib.org).
• We compare the time and storage requirements of
  the novel H-QR method with the respective
  quantities for
• the classic, full QR method, and
• a sparse, implicit variant called SPQR.

• Method Description
• Hierarchical Matrices were introduced in 1999 as
  a means of storing and manipulating large dense
  matrices. H-matrices are nice because they
  offer efficient storage and efficient
  (approximate) matrix operations such as matrix
  evaluations, sums, products, inverses, and
  factorizations.
• An H-matrix is divided into blocks, and each of
  these blocks, if suitable, is replaced with a
  low-rank representation computed by a SVD
  (singular value decomposition). Blocks which are
  not suitable are stored in their original form,
  as full matrices. Matrix arithmetic is performed
  with respect to these blocks with truncations to
  a lower rank wherever necessary.

• Objectives
• The objective of this undergraduate research
  project is to develop a new solver for saddle
  point systems through a combination of
• the well-known null space method, and
• the novel technique of hierarchical matrices.

Conclusions Through the development of our new
technique, we have been able to dramatically
reduce both the time and memory requirements to
compute a null basis Z which will accelerate the
numerical simulation of fluid flow. We will next
investigate the required accuracy of a null basis
and suitable preconditioners for the null space
method. With a successful outcome, more
accurate flow simulations will be feasible in
shorter time.
Acknowledgments This material is based upon
work supported by the National Science Foundation
under Grant No. 0408950 and by the Department of
Energy under DOE Grant No. DE-FG02-04ER25649. Any
opinions, findings, conclusions, or
recommendations expressed in this material are
those of the authors and do not necessarily
reflect the views of the National Science
Foundation or the Department of Energy.