TR3D:
3D optically thick NLTE radiative transfer

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Abstract

Transfer part Rate equation part Visualization

History References Versions

Abstract

Nnode * ( Nnue + Nlev ) reals,

Transfer part

In TR3D the radiative transfer equation for non-relativistic, stationary, frequency decoupled, and unpolarized radiative transfer is solved by a mean intensity approach.



First, the radiative transfer equation has to be discretized. TR3D uses an equidistant Cartesian mesh and first order upwind finite differences for the advection term.



To account for the direction dependence of the specific intensities the unite sphere in 3D space has to be discretized, discrete ordinates are introduced. This ordinate discretization is equidistant in longitude as well as in latitude, except that some ordinates are omitted towards the poles.



Once discretized, we can proceed to the actual heart of TR3D, the mean intensity approach. The basic idea for this approach stems from Turek. The point is to rewrite the discretized transfer equation in such a way that it becomes an equation for the discretized mean intensity J rather than for the directed (or specific) intensities I.



One ends up with a set of linear system of equations, AJ=E, one such set per discrete frequency to be treated. Here J is a vector containing the mean intensity for one particular frequency at each grid point, E is a vector containing, in essence, emission terms with the exception of scattering, and A is a rather full, non-symmetric, definite matrix, which is given only implicitly. This system of equations we solve using a BiCGStab algorithm with diagonal preconditioning.

Note that the above rewriting of the transfer equation requires an inversion of the matrix Thm. Such an inversion is efficiently possible as, for each discrete ordinate direction separately, the spatial grid can be renumbered such that the matrix Thm becomes lower triangular.

Rate equation part

The rate equations or statistical equilibrium equations, along with some conservations laws, describe the level populations of the different atoms and ions. Actually a non-linear system of ordinary differential equations, they are treated in TR3D as a non-linear system of equations only. This corresponds to the assumption that the time scales governing the rate equations are much faster than any flow time scale.



With the atomic processes considered within TR3D this system of equations has block structure and is non-linear in the electron density only.



Usually, the system is badly conditioned and the entries of the solution vector, the atomic level populations, easily differ by ten orders of magnitude.

The system is solved by iteration, using the values of the previous iteration step to get rid of the non-linearities in the present iteration step. For the solution of the then linearized system of equations Housholder transformations are used. Optically thick lines are taken into account using Sobolev coefficients, adapted to 3D.

Visualization

History

• 1993: Idea of generalized mean intensities brought up by Turek in the frame of LTE continuum transfer
• 1998: TR3D developed by Doris Folini in the frame of her PhD thesis
• 2002: Alexandre Desboeufs and Michael Psarros equip TR3D with an adaptive mesh

References

Available versions