1. Diffusion Model Error Assessment Jim E. Morel Texas A&M University CRASH Annual Review October 29, 2010

2. Outline The CRASH-like model problem. Diffusion versus transport modeling. Estimating diffusion model error with PDT alone. Future work.

3. Model Problem We have develop a CRASH-like model problem to investigate diffusion and transport issues in standalone calculations. This is useful in general, but absolutely essential given the fact that we cannot couple our PDT Sn transport code to the BATSRUS hydrodynamics code due to UCNI regulations.

4. Model Problem Gold Beryllium Plastic Shocked Source Region Post- Shock Xe Pre- Shock Xe Plastic

5. Problem Definition X-Y geometry Fully-implicit time discretization Lumped PWLD spatial discretization 1eV Planckian on all boundaries Te and Trad initialized to 1ev throughout the problem Constant electron source in shocked Xe with intensity 4.25E+33 eV/cm^3-s

6. Problem Definition Material properties: o Representative densities and specific heats chosen from CRASH simulations. o Opacities generated with the CRASH opacity code.

7. Diffusion versus Transport Modeling The CRASH problem has optically-thin Xenon within an optically thick plastic tube that has radiation incident at grazing angles at points far from the shock. Radiation incident upon an optically-thick diffusive region at grazing angles generates radiation boundary layers. Such boundary layers affect the rate at which energy enters the diffusion region. Diffusion theory cannot represent boundary layers.

8. Diffusion versus Transport Modeling The ablation of the tube wall generates a wall shock that strongly interacts with the main shock. Thus it is conceivable that the diffusion model could introduce significant error in the overall simulation. Simply comparing results from separate diffusion and transport packages is not as informative as one might think. Many factors can cause differences in the two solution other than the adequacy of diffusion theory.

9. Diffusion Versus Transport Modeling For instance, diffusion and transport packages can use different radiation-material coupling schemes, different spatial discretization schemes, different levels of nonlinear iterative convergence, etc. To overcome this difficulty, we have developed a new approach to obtain a measure of model error relative to the Sn model via Sn calculations alone.

10. The Corrective Diffusion Source We first solve the Sn equations over a time step to obtain the following balance equation: Next we add to both sides of the previous equation to obtain

11. The Corrective Diffusion Source where Note that Q represents the source that would have to be added to the diffusion equation to obtain the transport solution. Also note that Q contains boundary surface terms as well.

12. The Corrective Diffusion Source As such, Q is a quantitative indicator of diffusion model error. However, if Q is computed numerically, it will be a “pure” indicator of diffusion model error only if the diffusion discretization is fully consistent with the Sn discretization. To compute the actual error, one must solve the Sn- consistent diffusion equation without the corrective source and subtract that solution from the Sn solution.

13. Sn-Consistent Diffusion It is fairly straightforward to derive the Sn-consistent diffusion equation, so generating the corrective source is also straightforward. However, the Sn-consistent diffusion equation takes the form of an equivalent first-order system (energy- balance and Fick’s law) that can be notoriously difficult to solve relative to standard diffusion discretizations. Thus, while we will be able to compute a consistent corrective source, it may be difficult to compute a consistent diffusion model error.

14. Assessment of Consistency We can estimate the consistency of any diffusion discretization by computing the diffusion solution with the corrective source included. A consistent diffusion solution will equal the Sn solution. Thus the agreement between the diffusion and transport solutions is a measure of the consistency of the diffusion discretization.

15. Future Work We will investigate various ways to compute a “relative” corrective source. We will investigate the use of approximately-consistent Sn diffusion discretizations and check their consistency by using them in conjunction with the corrective diffusion source to compute the transport solution. All of this work will feed into overall assessment of diffusion model error for the CRASH experiment.