Numerical implementation of anisotropic diffusion

Abstract:  Modeling anisotropic diffusion is essential for various applications such as thermal transport in magnetized plasmas and image processing. I shall show that simple centered differencing of the anisotropic diffusion operator can lead to unphysical accentuation of temperature extrema (temperature can become smaller than the initial minimum!). The use of limiters (similar to those used in the reconstruction step of finite volume methods) can prevent this. Since explicit schemes require a small stability timestep, I shall present a semi-implicit generalization of our explicit method which is stable for longer timesteps and leads to a tridiagonal system of equations. I shall also present our numerical experiments with super-time-stepping, in which a much larger timestep is taken by wisely choosing the substep intervals, resulting a large speedup compared to the explicit method. The key advantage of super-time-stepping is that, unlike the semi-implicit method, it is straightforward to parallelize on massively parallel distributed memory clusters.