An Affine Semi-Lagrangian Advection Method

Abstract

In computer graphics, the standard semi-Lagrangian advection as in the work of Stam (1999) is a widespread unconditionally stable transport scheme used in incompressible fluid solvers. Due to its stability, which disconnects the grid resolution from the time step required to prevent the numerical solution from blowing up, the method provides a good artistic control over the quality-performance trade-off. However, it is also notoriously known to include a great amount of artificial dissipation into the solution, hence destroying fine-scale details, and making simulated fluids appear overly viscous.

Previous research efforts to counteract this unfortunate side effect have been spent notably on reinserting lost small-scale features, and on adapting different parts of the method to improve its accuracy. As part of the latter group, we present an affine semi-Lagrangian advection method, which we refer to as the ASLAM (pronounced "ay-slam"). This novel ASLAM adapts the locally affine descriptor of velocity from the affine particle-in-cell (APIC) method, a hybrid approach by Jiang et al. (2015), to the particle-free context of the Eulerian framework. We analyse the ASLAM's behaviour on a selection of testing scenarios, and evaluate it both qualitatively and quantitatively against a range of competing techniques, showing that it successfully reduces the artificial dissipation arising from standard semi-Lagrangian advection.