A numerical study of Burgers' equation.
This thesis is concerned with various numerical
approaches to the solution of the one dimensional form of
Because of its similarity to the Navier-Stokes
equation, Burgers' equation often arises in the mathematical
modelling used to solve problems in fluid dynamics involving
turbulence. Difficulties have been experienced in the past
in the numerical solution of this equation particularly for
small 1) (i.e. for large Reynolds number) which corresponds
to steep fronts in the propagation of dynamic waveforms.
This thesis describes a number of numerical approaches
including finite-difference methods and a Fourier series
approach which are shown to produce high accuracy for
large -\) (i.e. small Reynolds number) by comparing with
the analytical solution.
To obtain high accuracy for small ~ (i.e. large
Reynolds number) a finite element approach is necessary.
This method is described for the case of fixed nodes by
using cubic polynomials for the shape function. Improvements
can be obtained by choosing the size of the elements to take
into account the nature of the solution. The aim is to
"chase the peak" by altering the size of the elements at
each stage using information from the previous step. This
moving node approach is described in the thesis.
Another possible numerical technique discussed is the
use of a variational-iterative scheme based on complementary
variational principles. This has been successfully applied
to the steady state case of Burgers' equation and an
extension to the full Burgers' equation is described. For
further comparison purpose"s resul ts are obtained by using
a cubic spline collocation method.
Results have been obtained by applying all these
numerical techniques to Burgers' equation under specified
boundary and initial conditions. These results have been
discussed and, where possible, compared to the analytical