Active flux schemes on moving meshes with applications to geometric optics. (English) Zbl 07785513
Summary: 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.
MSC:
| 65Mxx | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 76Mxx | Basic methods in fluid mechanics |
| 35Lxx | Hyperbolic equations and hyperbolic systems |
Keywords:
Liouville’s equation; active flux scheme; geometric optics; hyperbolic conservation law; moving meshSoftware:
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