Super-time-stepping for advection-diffusion

In this thesis, we test a method known as Super-time-stepping (STS). It is a method for speeding-up explicit numerical time-stepping schemes for parabolic partial differential equations, by relaxing the stability restriction on the time-step.

We apply STS on an advection-diffusion problem, for various ratios of velocities to diffusivities (Peclet numbers), spanning from pure diffusion to advection dominated. We compare the preformance of the STS scheme with some standard implicit schemes (Crank-Nicolson and Fully Implicit) using both direct (tridiagonal) and iterative (SOR) linear system solvers, as well as with the standard explicit scheme. Our model advection-diffusion problem admits exact solution, and thus we calculate the errors to gauge accuracy, along with speed of execution.

We find that STS is easy to implement on an existing explicit code, and it accelerates the standard explicit scheme considerably. On diffusion-dominated and mildly advective problems, STS is more efficient than the implicit scheme with the iterative SOR solver, but on highly advective problems the fully implicit/SOR scheme was faster. Implicit schemes using the tridiagonal solver were more efficient than STS for our one-dimensional linear problems, but the Tridiagonal Algorithm is inapplicable when solving two or three dimensional problems or when solving non-linear problems.