A Mixed Eulerian–Lagrangian scheme for scalar transport

The tracking of pollutants in gas and liquid is a major problem to address in atmospheric environment, air quality characterization, and industrial material processes. A Lagrangian scheme devoted to the approximation of the advection term in an advection–diffusion equation is proposed to deal with small diffusivity values or large Péclet numbers. The Lagrangian scheme is used in practice as an Eulerian method discretized on Lagrangian marker points. Advection and diffusion of a circular concentration spot in a vortex flow are considered for validation purpose. The resulting mixed Eulerian–Lagrangian scheme reduces the numerical diffusion to almost computer error and provides better results than other Eulerian classical schemes of the literature. The scheme is finally illustrated in a natural convection situation.