Active flux schemes on moving meshes with applications to geometric optics

• Main Authors: Bart S. van Lith, Jan H.M. ten Thije Boonkkamp, Wilbert L. IJzerman
• Format: Article
• Language: English
• Published: Elsevier 2019-06-01
• Series: Journal of Computational Physics: X
• Online Access: http://www.sciencedirect.com/science/article/pii/S2590055219300460

Active flux schemes are finite volume schemes that keep track of both point values and averages. The point values are updated using a semi-Lagrangian step, making active flux schemes highly suitable for geometric optics problems on phase space, i.e., to solve Liouville's equation. We use a semi-discrete version of the active flux scheme. Curved optics lead to moving boundaries in phase space. Therefore, we introduce a novel way of defining the active flux scheme on moving meshes. We show, using scaling arguments as well as numerical experiments, that our scheme outperforms the current industry standard, ray tracing. It has higher accuracy as well as a more favourable time scaling. The numerical experiments demonstrate that the active flux scheme is orders of magnitude more accurate and faster than ray tracing. Keywords: Liouville's equation, Active flux scheme, Geometric optics, Hyperbolic conservation law, Moving mesh