The numerical solution of convection--diffusion--reaction equations is required in many applications, e.g. in the simulation of a chemical precipitation process, which is modeled via a population balance system. We are especially interested in numerical simulations of droplet size distributions in clouds whose behavior can also be described with the help of a population balance system consisting of the Navier--Stokes--equations and a further convection--dominated equation.
In coupled systems, an inaccurate solution of this equation may easily lead to instable simulations, in particular if the computed solution possess spurious oscillations. The use of an accurate oscillation--free method is therefore crucial. We have studied several different stabilized finite element methods for convection--dominated problems on the basis of scalar time--dependent convection--diffusion--reaction equations with small diffusion. We compared the methods with regard to the following factors: They must be able to compute sharp layers and the solutions must not possess spurious oscillations. Among the studied schemes were the Streamline--Upwind Petrov--Galerkin (SUPG) method, several Spurious Oscillations at Layers Diminishing (SOLD) methods and a local projection stabilization (LPS) scheme. We also tested two Finite Element Methods Flux--Corrected--Transport (FEM-FCT) schemes.
The methods will be assessed with respect to the criteria mentioned above.