Conserved and primitive variables: Conversion and inversion

Conserved and primitive variables

Conserved variables are first discussed in this section. For the equations of ideal GRMHD the conserved variables vector is,

.

In addition to these, fluid codes define an additional vector of variables called primitives (P). These variables have the following advantages,

• Facilitate computing the interface states via reconstruction and consequently the fluxes.
• Facilitate computing the fluid source terms (since it involves computing the components of the stress-energy tensor)
• Can be physically better understood.

The primitive variables vector is,

.

where . vⁱ is defined here. This was done to increase numerical stability.

Primitives ---> conserved variables (conversion)

The conserved variables are analytic functions of the primitives and can be computed exactly. This is done in two steps,

1. The fluid four velocity (u^µ) and magnetic induction vector (b^µ) are computed from the primitives.

,

,

,

,

where Equations (1) and (2) are true because the fluid four velocity is measured w.r.t. a normal observer. Equation (3) is from the definition of b^µ, while Equation (4) can be derived from , which is true in the ideal MHD limit. The function get_state_vec found in phys.c computes the four vectors.

2. The components of the stress-energy tensor are calculated from the primitives and the four vectors. The conserved variables are evaluated from these.

Conserved variables ---> primitives (inversion)

The matter conserved variables are nonlinear functions of the corresponding primitives which have no known analytic inverse expressions. A ¹D_W scheme (Mignone & McKinney (2007) and Noble et al. (2006)) is used to numerically compute the matter primitives. The magnetic field primitives are equal to the corresponding conserved variables up to a factor of . iharm inverts the conserved variables twice during a single time step: (i) following the update to U in advance_fluid having computed fluxes and source terms (ii) following the application of fluid floors in fixup_floor.

The inversion scheme is not guaranteed to successfully converge to reasonable values for the primitives in a finite number of iterations. iharm fixes the 'bad' values of the inverted primitives by interpolating from nearby zones (fixup_utoprim).