This page shows the mathematical forms of the governing equation pieces that each kernel class represents. Note that \(u\) is used in some places to denote a generic variable. Elsewhere typical literature symbols are used, e.g. \(\phi\) denotes the neutron flux, \(T\) denotes the temperature, and \(C\) denotes precursor concentrations.

CoupledFissionEigenKernel

\[-\frac{\chi_g^p}{k} \sum_{g' = 1}^G (1 - \beta) \nu \Sigma_{g'}^f \phi_{g'}\]

CoupledFissionKernel

\[-\chi_g^p \sum_{g' = 1}^G (1 - \beta) \nu \Sigma_{g'}^f \phi_{g'}\]

CoupledScalarAdvection

\[\nabla \cdot \vec{a} u\]

DelayedNeutronSource

\[-\chi_g^d \sum_i^I \lambda_i C_i\]

DivFreeCoupledScalarAdvection

\[\vec{a} \cdot \nabla u\]

FissionHeatSource

\[-\frac{P}{\int_{\partial V} \sum_{g' = 1}^G \nu \Sigma_{g'}^f \phi_{g'} dV} \sum_{g' = 1}^G \nu \Sigma_{g'}^f \phi_{g'}\]

where \(P\) is a representation of the total power of the reactor.

GammaHeatSource

\[-\gamma Q_f\]

where \(\gamma\) is a factor representing heat dissipation by gamma and neutron irradiation in the moderator and \(Q_f\) is given by:

\[\sum_{g=1}^G \epsilon_{f,g}\Sigma_{f,g}\phi_g\]

with \(\epsilon_{f,g}\) the amount of heat given off per fission event.

GroupDiffusion

\[- \nabla \cdot D_g \nabla \phi_g\]

InScatter

\[-\sum_{g \ne g'}^G \Sigma_{g'\rightarrow g}^s \phi_{g'}\]

NtTimeDerivative

\[\frac{1}{v_g}\frac{\partial \phi_g}{\partial t}\]

PrecursorDecay

\[\lambda_i C_i\]

PrecursorSource

\[-\sum_{g'= 1}^G \beta_i \nu \Sigma_{g'}^f \phi_{g'}\]

ScalarAdvectionArtDiff

\[\nabla \cdot -\delta \nabla u\]

where \(\delta\) is an artificial diffusion coefficient determined by:

\[\delta = \frac{|\vec{a}| h_{max}}{2}\]

with \(\vec{a}\) the advection velocity and \(h_{max}\) the maximum element length dimension.

ScalarTransportTimeDerivative

\[\frac{\partial u}{\partial t}\]

SelfFissionEigenKernel

\[\frac{-\nu_f \Sigma_f \phi}{k}\]

SigmaR

\[\Sigma_g^r \phi_g\]

TransientFissionHeatSource

\[-\sum_{g=1}^G \epsilon_{f,g}\Sigma_{f,g}\phi_g\]