Fluxes
HallThruster.jl uses the Finite Volume method, and as such the face values of the fluxes need to be reconstructed. See Numerics for more information.
The fluxes $F_{_{i+\frac{1}{2}}}$ and $F_{_{i-\frac{1}{2}}}$ are reconstructed at the cell interfaces, and for this flux reconstruction multiple options are available. These are set using the object HyperbolicScheme consisting of fields flux, limiter, and reconstruct. Three different flux approximations are available.
| Flux | Description |
|---|---|
upwind | Simple first order accurate flux approximation, that as a results does not distinguish between cell centered and cell average values and adapts reconstruction according to sign of advection velocity. Very diffusive. No Riemann solver or approximation. |
HLLE | Approximate Riemann solver. The Harten-Lax-van Leer-Einfeldt scheme approximates a Riemann problem with three constant states. see reference. The scheme is positively-conservative if stability bounds for maximum and minimum wavespeeds are met, which makes it useful in its application with HallThruster.jl. First order accurate in space. B. Einfeldt. On godunov-type methods for gas dynamics. Journal of Computational Physics, 25:294-318, 1988. |
rusanov | Approximate Riemann solver. Also known as the local Lax-Friedrich flux. Has slighlty modified choice of wave speeds. Adds viscosity to a centered flux. More diffusive than HLLE. Chi-Wang Shu, Lecture Notes: Numerical Methods for Hyperbolic Conservation Laws (AM257) |
These flux approximations are all first order accurate in space (piecewise constant recontruction), but can be extended to piecewise linear reconstruction within a cell. To satisfy stability bounds and keep the scheme total variation diminishing (TVD), it has to be coupled with a limiter. Many limiters have been proposed, the ones implemented in HallThruster.jl are the following: koren, minmod, osher, van_albada, van_leer. If the field reconstruction is set to true, the selected limiter will be used.