# Symbolic Implicit Monte Carlo

## The Origins of Implicit Monte Carlo

A flexible and accurate method for solving nonlinear, frequency-dependent radiation transport problems called Implicit Monte Carlo (IMC) was first introduced in early 1971 by Lawrence Livermore scientists J. A. Fleck, Jr. and J. D. Cummings.

Pros:

• The method has been used successfully to solve non-equilibrium, line transport problems as well as local thermodynamic equillibrium (LTE) problems.
• Extremely robust
Cons:
• The method is computationally expensive in materials that are highly opaque.
• IMC has poor time step convergence characteristics by behaving like SIMC as the time steps are shrunk.

## Symbolic Implicit Monte Carlo

In Symbolic Implicit Monte Carlo (SIMC) photon bundles carry symbolic weights until census is reached and the calculation time is independent of the opacity of the medium. Originally introduced by Eugene Brooks, SIMC obtains excellent computational efficiency and numerical stability for optically thick media. This introduced a new implicit Monte Carlo technique for solving time dependent radiation transport problems involving spontaneous emission.  In this technique, the effective scattering term does not occur and the execution time for the Monte Carlo portion of the algorithm is independent of opacity.

Pros:

• SIMC is computationally efficient at high opacity because time to solution is independent of the opacity of the material through which the photons travel.
Cons:
• Requires a solution of a system of linear equations at the end of each time step
• Based on how the system is discretized, photons may appear to travel too fast since they travel in a straight line. IMC's effective scattering masks this effect.

## Future Prospects

Currently, work is underway to further improve the computational efficiency of SIMC.  Eugene Brooks, Frank Daffin, Scott McKinley and Abraham Szoke have been investigating the effects of frequency biasing as a means of variance reduction in the calculation.  Biasing in space and time is under investigation as well.

A Fully Implicit version of SIMC has been performed and analysized. It only requires a little additional documentation for it to be published. The Fully Implicit version is suitable for laser work where high gain makes SIMC and IMC unstable.

In addition, work has begun to derive a Difference Formulation that is based on solving the difference between the photon flux and the quasi-steady-state solution. Results in this are too early to state. However, it appears that the formulation is highly efficient and accurate in very opaque regions where the temperature changes gradually.

 UCRL-WEB-200195 mckinley9@llnl.gov Updated:  October 9, 2003