Stochastic Implicit Euler burnup scheme

The Stochastic Implicit Euler (SIE) burnup scheme is an alternative burnup scheme that can be used if the traditional predictor-corrector burnup schemes in Serpent exhibit unstable behavior.

Background

The theoretical background of the SIE scheme is described in

Input

The SIE burnup scheme can be chosen with the set sie input card. The number of iterations for each burnup steps can be specified.

Some additional input options that many times are combined with the set sie card are set fsp for passing the fission source from the end of previous neutron transport solution to the beginning of the next one, and set cpop to use an alternate (typically lower) neutron population for the SIE iteration (compared to the neutron population used on the initial predictor step.

Implementation

For burnup calculations, Serpent needs to evaluate the one group flux $\phi$ for each depletion region and one group fission/transmutation cross sections $\Sigma$ for each fission/transmutation reaction of each nuclide.

These can then be used for calculating the reaction rates required for assembling the burnup matrix $\mathbb{M}$. The following algorithmic description uses $\phi$ to refer to both the one group burn flux and the one group cross sections.

1:	input: $\mathbf{N}_{0}$			Start from initial nuclide concentrations.
2:	$\phi_{0} \leftarrow \phi_{B}(\mathbf{N}_{,0})$			Calculate $\phi$ using initial nuclide concentrations.
3:	for $i \leftarrow 0,1,\ldots$ do			Loop over burnup points.
4:		$\mathbf{N}_{i+1}^{(0)} \leftarrow \exp \left[ \mathbb{M}(\phi_{i}) \Delta t\right] \mathbf{N}_{i}$		Burn forward (predictor) using previously obtained BOS $\phi$.
5:		for $n \leftarrow 1,2,\ldots, c$ do		Loop over iterations for current burnup point.
6:			$\phi^{(n)}_{i+1} \leftarrow \phi_{B} (\mathbf{N}^{(n-1)}_{i+1})$
	xx	x

Noteworthy things

• No neutron transport solution is actually calculated using the final concentrations for a certain point.