Convection in a PDE

In my last two blog posts I tried to explain the mechanism of convection and presented some examples of convection in nature. The starting point has been the post about Interest Rate Models - From SDE to PDE where we ended with a diffusion - convection - reaction equation. The numerical solution using standard discretisation methods entails severe problems, resulting in strong oscillations in the computed values. The drift term (first derivative) is chiefly responsible for these diculties and forces us to use specif- ically developed methods with so-called upwind strategies in order to obtain a stable solution. Very roughly speaking, it is mandatory to “follow the direction of information flow”, and to use information only from those points where the information came from. In trinomial tree methods, the up-branching and down-branching takes into account the upwinding and leads to nonnegative weights which correspond to stability.

Left: Trinomial trees cut off the extreme ends to avoid stability problems: Change of calculation boundary!
Right: Flow of Information - If convection plays an important role, upwinding has to be applied.

Originating from the field of computational fluid dynamics, upwind schemes denote a class of numerical discretisation methods for solving hyperbolic partial differential equations. In our case, they are used to cure the oscillations induced by using standard discretisation techniques in convection dominated domains of combined diffusion-convection-reaction equations. Upwind schemes use an adaptive or solution-sensitive finite difference stencil to numerically simulate the direction of propagation of information in a flow field in a proper way. Starting with next weeks blog post I will show in detail how unwinding schemes work and how we can use them to cure instabilities induced by convection.