Space-Time Spectral Element Methods for One-Dimensional Nonlinear Advection-Diffusion Problems

The following space-time Galerkin spectral element methods are developed and applied to solve the Burgers equation with small viscosity: (a) coupled methods, consisting of an explicit method for hyperbolic dominated equations and an implicit method for parabolic dominated equations; (b) two splitting methods which solve the hyperbolic substep explicitly and the parabolic one implicitly (one uses spectral elements in the explicit part and the other uses the Adams-Bashforth multistep method). A subcycling technique, in which several convective steps are taken for each implicit viscous step was also investigated for the two splitting methods. A stability analysis of the four methods is performed and subsequent results are debated. A convergence study and a comparison of computer execution time for the four methods is made and the results are discussed. Comparative study leads to the conclusion that the space-time spectral element splitting method with subcycling is superior to the other methods presented in terms of robustness and computer execution time. The number of subcycles should be kept low (2-3) in order to avoid significant loss of accuracy. The coupled explicit method is also applied to the solution of the one-dimensional coupled continuity, momentum, and energy equations for non-isothermal flow of an ideal gas with temperature dependent properties in a cylindrical duct of variable radius.