Lagrangian Methods and Finite Elements in Shallow Water Modelling

by
Mr Mark Trew and
Associate Professor Michael O'Sullivan

Department of Engineering Science 
School of Engineering
The University of Auckland
Private Bag 92019 
Auckland
New Zealand

Telephone: 64 09 373 7599 x 8382
Fax: 64 09 373 7468
email: trew@esv1.auckland.ac.nz

Abstract:

Advective transport processes are significant in many shallow water flow
problems and must therefore be appropriately modelled. This is particularly
the case when computed hydrodynamic results are to be used as the basis of
contaminant transport modelling.

Traditionally the advective terms of the momentum conservation equation,
non-self-adjoint first order terms, have posed significant numerical
challenges to solving the shallow water equations. Stable solutions have been
obtained through the use of eddy viscosity terms.  The amount of eddy
viscosity required, often based on the element size, is generally found to be
physically excessive. Many methods presented in the literature use these
approaches and they are highly developed. However, their excessive dissipation
may be detrimental to resolving some of the features of advective flows.

A non-linear semi-Lagrangian operator splitting method is proposed where
principally the advective operators are split from the remaining operators in
the shallow water equations and are dealt with separately. This split operator
approach models advection by integrating along streamlines in the flow. The
remaining operators are then integrated over the domain of interest using a
Galerkin finite element method. Excessive eddy viscosity is not necessary for
maintaining stability. Algorithms that minimise computation while performing
streamline integration through unstructured computational meshes are
developed. Large systems of non-linear sparse equations are formed and solved.

The model is applied to a case study of shallow water flow in New Zealand. The
results form a basis for subsequent contaminant transport modelling in the
region.


Key Words:

Environmental Problems (primary topic), Computational Fluid Dynamics
(secondary topic one), Finite Element Methods (secondary topic two).





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