## Higher-Order and Multi-Level Time Integration of Stochastic Differential Equations and Application to Coulomb Collisions

#### Andris DimitsLawrence Livermore National Laboratory

We have examined the extension of Langevin-equation Monte-Carlo algorithms for Coulomb
collisions from the conventional Euler-Maruyama time integration to the next higher orders of accuracy, as
well as multi-level schemes based on these. This examination has yielded many results, some of interest
specifically for the Coulomb-collision problem, some of fundamental importance for integration of
stochastic motion on a spherical (and, more generally, any curved) surface, and some of considerable
utility for integration of multi-dimensional systems of stochastic differential equations (SDE’s).

In one common Langevin-equation approach, the angular scattering is treated with a combination
of near-Cartesian stochastic velocity-direction kicks, in a unit-vector frame that is rotated so that at the
beginning of each timestep the third axis is aligned with the velocity direction. We find that using this
approach, the angular component of the collisional scattering can not be extended to give better strong
convergence than that for the Euler-Maruyama scheme. Instead, the extension to a higher order (Milstein)
scheme proceeds via a formulation of the angular scattering directly as SDE’s in the two fixed-frame
spherical-coordinate variables. This extension has been implemented, and results have been obtained for
Coulomb collisions showing the improved strong convergence.

The resulting Milstein scheme is of value both as a step towards algorithms with improved
accuracy and efficiency either directly through algorithms with improved convergence in the averages
(weak convergence) or as a building block for multi-time-level schemes. The latter have been shown to
give greatly reduced cost for a given overall error level compared with conventional Monte-Carlo
schemes, and their performance is improved considerably when the Milstein algorithm is used for the
underlying time advance versus the Euler-Maruyama algorithm. Progress in these directions for the
Coulomb collision problem will be reported.

In collaboration with M.S. Rosin, L. Ricketson and R.E. Caflisch, UCLA, and B.I. Cohen, LLNL.
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* Work performed for US DOE by LLNL under Contract DE-AC52-07NA27344 and by UCLA under
grant DE-FG02-05ER-25710.

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