Applied Mathematics Seminar

In many physical problems, there are several competing processes which often have a distinct character. For example, in compressible flow there are sound waves and incompressible, vortical motions. Most numerical methods for such problems use variables that do not distinguish between these processes. For example, using the velocity as a variable in compressible flow mixes the sound waves and the vortical motions, one which is fast and the other which is relatively slow. We show, via an example in geophysical fluid dynamics, analogous to compressible flow, that distinguishing these processes via the initial choice of variables used in the numerical simulation, can greatly improve accuracy and computational efficiency. This improvement is normally much greater than that coming from the details of the numerical discretisation approach. Specifically, there appears to be an optimal choice of variables for a given problem, or flow regime. We discuss an idea for adapting the variables used dynamically, both in space and in time, to achieve the most optimal numerical simulation.