A numerical study of Burgers' equation

• Main Author: Wanless, P.
• Published: University of Sunderland 1984
• Subjects: 510; Pure mathematics
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.349102

This thesis is concerned with various numerical approaches to the solution of the one dimensional form of Burgers' equation. 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 results.