# numericalFlux¶

The numericalFlux attribute is used to specify the method of computing an upwind flux at the cell interface. The precise details of the mathematics and algorithm used is outside the scope of this document; further information on the mathematical formulation can be found at https://en.wikipedia.org/wiki/Riemann_problem and details of the algorithm at https://en.wikipedia.org/wiki/Riemann_solver.

The recommended choices for the numericalFlux attribute are:

hlleFlux (available for all equation systems; use with caution)

Provides a diffusive, but robust estimate of the flux at the interface. Appropriate for high mach number flow problems, but excessively diffusive for problems where intermediate waves are important (e.g. contact discontinuities). Because of the high level of diffusivity associated with hlleFlux, caution should be used interpreting the results. The USim implementation of the hlleFlux is:

\notag \begin{align} \tilde{f}_{i+1/2}=\frac{\left(\lambda_{min}\,f^{-}_{i+1}-\lambda_{max}\,f^{+}_{i}\right)}{\left(\lambda_{max}-\lambda_{min}\right)}+\frac{\lambda_{max}\lambda_{min}}{\lambda_{max}-\lambda_{min}}\left(q^{-}_{i+1}-q^{+}_{i}\right) \end{align}

Where $$\lambda_{max}$$ is the maximum wave speed of the system at the interface, and $$\lambda_{min}$$ is the minimum wave speed of the system at the interface.

hllcFlux (eulerEqn only; recommended for neutral flows if roeFlux is not sufficiently robust)
Incorporates physics due to contact discontinuities within the flux. This approach provides greater physics fidelity than the hlleFlux but this comes at the cost of reduced robustness.
hlldFlux (mhdDednerEqn only; recommended for ionized flows if roeFlux is not sufficiently robust)
Incorporates physics due to MHD waves within the flux. This approach provides greater physics fidelity than the hlleFlux but this comes at the cost of reduced robustness.
roeFlux (Recommended choice for neutral and ionized flows)
Implements Roe’s exact solution of the linearized Jacobian for eulerEqn, mhdDednerEqn and related equations (see https://en.wikipedia.org/wiki/Roe_solver for futher details). May not be sufficiently robust for flows with strong rarefactions, in which case hllcFlux, hlldFlux should be considered.

In addition, USim provides a range of numericalFlux options, designed for expert users.