I began to work on this project in 2007 in close collaboration with Joachim Maruhn (Frankfurt University, Frankfurt am Main, Germany) and Anna Tauschwitz (GSI, Darmstadt, Germany). The primary motivation for all three of us was the necessity to develop an adequate theoretical tool for modeling high-temperature laser plasmas, generated in the experiments that are being conducted at GSI with the NHELIX and PHELIX lasers. It is well known that the dynamics of such laser-created plasmas is to a large extent dominated by radiative processes. Other important potential applications include simulations of strongly radiating plasmas in multi-wire Z-pinches, and simulations of indirect-drive ICF targets.
Usually radiation transport in hydrodynamic codes is described in a diffusion approximation for each spectral group. My collaborators and I have set up an ambitious goal to develop a two-dimensional (2D) hydrodynamics code coupled with the solution of a quasi-stationary transport equation for the frequency-dependent radiation intensity. Very few such codes exist in the world, and numerical methods for combining radiation transport with 2D and 3D hydrodynamics are not well developed. In our effort we profited substantially from the previous work by R.Ramis and his co-authors .
Fortunately, in our work we did not have to start from scratch and could build upon a pure 2D hydrodynamics package CAVEAT , developed about 20 years ago at Los Alamos. The CAVEAT code has been based on a second-order Godunov-like scheme for a structured quadrilateral grid, and fully exploits the advantages of the arbitrary Lagrangian-Eulerian (ALE) technique. Its numerical scheme produces just the right amount of numerical diffusion, so that, for example, in the Rayleigh-Taylor unstable situations the shortest (on the scale of a single grid cell) wavelengths are conveniently suppressed, whereas at wavelengths of 8-10 grid cells the instability is quite accurately reproduced . An important advantage of the CAVEAT numerical scheme for hydrodynamics is that it is able to maintain very accurately the inherent symmetry of the problem (due to the structured character of the mesh and moderate amount of numerical diffusion) and, for example, follow symmetric implosions down to the radial convergence factors of ~100 .
As a first step, we had to incorporate into the CAVEAT package an efficient algorithm for non-linear thermal conduction. We based our scheme on the symmetric semi-implicit (SSI) method , which appears to have no alternative if, as the next step, the multi-dimensional hydrodynamics is to be efficiently coupled with numerical solution of the radiation transfer equation. Our algorithm for thermal conduction is described in detail Refs.  and . It has been thoroughly tested and proved to be quite accurate and efficient, especially when the ALE technique is used to avoid strong mesh deformations.
Development of a reliable 2D scheme for radiation transport, which could be efficiently coupled with hydrodynamics, proved to be a more challenging task. One of the major difficulties is the requirement that the numerical scheme for solving the transfer equation should automatically reproduce the solution of the diffusion equation (the diffusion limit problem) in the limit of large optical thickness. Nevertheless, by the end of 2010 we had a working version of the new code - which was named RALEF-2D (Radiation Arbitrary Lagrangian-Eulerian Fluid dynamics) - but only in the Cartesian (x,y) geometry. In 2013 the newly developed (so far unpublished) numerical algorithm was extended to the axi-symmetric r-z geometry. Main features of the numerical scheme and the results of several tests are described in the unfinished (the work is in progress) report . As an example, the linked Figure illustrates simulation results, obtained with the RALEF-2D code for a half-a-micron thin carbon foil, irradiated by a laser beam (irradiation intensity 0.5 TW/cm^2, laser beam propagates horizontally from right to left) with a strongly non-uniform intensity distribution across the focal spot.
We discretize the radiation transfer equation over photon propagation directions by using the classical Sn method, for which we need the nodes and weights of the angular quadrature formula on the surface of a unit sphere. The Sn nodes provided in the book by Carlson  are by far not sufficient for modern numerical simulations. In the RALEF code we use highly symmetric sets of the LSH angular ordinates and weights tabulated in Ref.  for n = 2, 4, 6, and a simpler and less symmetric ESn quadrature from Ref.  for n > 6; in the latter case there is a relatively simple algorithm for calculating the nodes , which all have equal weights. For n = 8, 10 and 12, for which both options are available, the preference is given to a simpler ESn quadrature because it tends to produce smaller spurious intensity fluctuations due to the ray effect.
A possible alternative would be to use a set of symmetric (with respect to the octahedron group with inversion) weights calculated by V.I.Lebedev, whose order ranges from n=9 to n=131 [12-16]. Unfortunately, the Lebedevs' nodes do not suit our purpose because they always have points on the (x,y,z) coordinate planes, whereas our scheme is based on Sn nodes that lie strictly inside each octant.
An introduction to the RALEF code can be found in the following presentation PDF file , prepared for a seminar at ILE (Osaka) on December 17, 2010.
The most recent presentation on RALEF in Prague at ELI Prague 2014-07-10 - pptx file is accompanied by a short movie Sns14-160.mp4 , which illustrates evaporation of an Sn microsphere by a 100-ns long pulse of CO2 laser, simulated in the r-z geometry.
Copyright © 2007 - 2014 Mikhail Basko
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Originally composed: 2009.06.27
Last update: 2014.07.31