A comparison of radiation transport and diffusion using PDT and the CRASH code Fall 2011 Review Eric S. Myra Wm. Daryl Hawkins

Our goal is to quantify error associated with using flux-limited diffusion in CRASH Key goals: Using PDT and CRASH, perform “method verification,” with the aim of improving the implementation of radiation diffusion and better understanding its shortcomings As necessary, perform code-to-code comparison and verification in the diffusion regime To the extent possible, set up the full CRASH problem in both codes and quantify the uncertainty of using diffusion vs. full transport 2 This study was recommended by the 2010 Review Committee PDT/CRASH coupling not presently an option

An objective comparison of transport and diffusion is challenging Differences in — discretization and solution methods — phase space coverage (full vs. a subset) — treatment of multiphysics coupling (e.g., matter-radiation energy exchange) Characterizing the effects of ad hoc features of a model — flux limiters in diffusion — use of microphysics (e.g., opacities) Procedural differences — e.g., the code may be used for a test problem in a different mode than for “real” problems (timestep selection, use of converged temperatures, etc.) A problem that’s easy for one code can be difficult for the other 3

Flux-limited diffusion approximates transport The full transport equation (used by PDT). The radiation energy equation (used by CRASH) is the zeroth angular moment of the transport equation with diffusive closure attained by Fick’s law. withand 4

Flux-limited diffusion approximates transport The full transport equation (used by PDT). The radiation energy equation (used by CRASH) is the zeroth angular moment of the transport equation with diffusive closure attained by Fick’s law. withand 5

Target problems determine how we use each code PDT: a deterministic radiation transport code Rad energy: gray and multigroup (both used) Rad angle: discrete ordinates (256 angles used) Spatial: discontinuous finite element method Time: fully implicit CRASH: an Eulerian rad-hydro, flux-limited-diffusion code Rad energy: gray and multigroup (both used) Rad angle: angle-averaged—0th angular moment equation, with 1st angular moment equation replaced by flux-limited diffusion Spatial: finite volume method Time: fully implicit 6

The starting point for comparison is diffusion-limit test problems Gray transport Simple opacities, but which may vary sharply across an interface Examples: — Infinite medium problems to test rates — Front problems to test wave propagation — Marshak waves to test propagation and rates — Added heat sources as a proxy for shock heating Concerns: — Choosing physically relevant timescales — Computationally tractable in a reasonable time by both codes — Defining “diffusive” for purposes of code comparison If done with care, the codes should agree closely 7

Both codes advance a diffusive front similarly Gray transport Uniform density of 1 g cm -3 Opacity = 10 5 cm 2 g -1 in strip Opacity = 10 4 cm 2 g -1 outside, but no emission-absorption TeTe T rad = 1 eV Initial conditions Results for radiation At t = 3.0 ps… Results for each code are virtually identical for T rad (PDT in maroon; CRASH in blue dashes) T e unchanged for both t diff ~ 10 ns, t fs ~ 3.0 ps, therefore diffusive 8

A Marshak wave with a heat source also agrees well Gray transport Uniform density of 1 g cm -3 Opacity = 10 5 cm 2 g -1 in strip Opacity = 10 3 cm 2 g -1 outside Emission-absorption active everywhere dQ/dt = 4.25 x eV cm -3 s -1 in central strip TeTe T rad = 1 eV Initial conditions At t = 100 ps, agreement is good Material energy transport matches Volume vs. surface effect? 9 PDT CRASH Q added

A more realistic test problem has been formulated 0.10 cm 0.08 cm 0.20 cm 0.02 cm 0.05 cm Plastic Au Be: higher opacity Post- shock Xe Pre-shock Xe: lower opacity Shocked Xe cm cm Hydrostatic 2D Cartesian No heat conduction Realistic opacities, using the CRASH tables A heat source acts as a proxy for shock heating T e = T rad = 1.0 eV, initially C v (Xe, Au) = 9.9 x eV g -1 K -1 C v (Be, Pl) = 1.1 x eV g -1 K -1 dQ/dt = 4.25 x eV cm -3 s -1 opacity cliff The heat source is active within this region 10

A 1D gray version of the problem provides a first look t = 2.0 ps t = 5.0 ps t = 20.0 ps t = 50.0 ps ____ CRASH FLD on _ _ _ CRASH FLD off ____ PDT Transport T rad shows only qualitative agreement on this problem t = 50.0 ps Material energy transport differs significantly 11

Agreement starts to improve in 1D multigroup comparisons ____ CRASH FLD on ____ PDT Transport T rad shows good agreement at early times, then starts to diverge Material energy transport still differs significantly. However, in multigroup, PDT now moves more energy, esp. upstream “Upstream” radiative pre- heating t = 2.0 ps t = 5.0 ps t = 20.0 ps t = 50.0 ps groups, geometrically spaced, 1.0 eV–20 keV

These results suggest some next steps 1D Xe-on-polyimide problem—relevant to wall ablation Complete the suite of runs using the 2D version of the CRASH setup Implement a second problem using snapshots from full-system CRASH rad-hydro runs as initial conditions. — Provides more realistic initial conditions (e.g., temperatures) — Mitigates initial transients and uncertainties in the appropriate timescale over which to make comparisons — Allows direct comparison between successive rad-hydro CRASH timesteps and PDT A preliminary 2D result using CRASH showing T rad 13

Conclusions We have constructed a test environment that allows comparison of radiation transport and diffusion for problems relative to the CRASH. PDT and CRASH show good agreement on a set of problems where they should agree. PDT and CRASH show a mixture of agreement and discrepancy for more realistic CRASH-relevant problems. Further study is warranted to determine if these discrepancies are significant for predictive simulations of the CRASH experiment. 14