Contents

1. About
2. Theoretical framework
3. Numerical schemes
4. Setting-up a simulation
5. Running a simulation
6. Example
7. Contributors
8. Publication list

About

Collective Oscillation Simulation Engine for Neutrinos – COSE\(\nu\) is written completely in C++ and provides two advanced numerical schemes to simulate collective neutrino oscillations in the mean-field limit. The first method uses fourth order central finite differencing supplimented by third order Kreiss-Oliger dissipation scheme. The second one is implemented with the finite volume method along with the seventh order weighted essentially non-oscillatory scheme for the flux reconstruction across the cell boundaries. In both cases the time evolution is carried out via fourth order Runge-kutta method.

Theoretical framework

COSE\(\nu\) solves the following 1-D hyperbolic equation with a source term which describes the evolution of a two-flavor neutrino system.

\[\begin{equation} \left( \frac{\partial}{\partial t} + \mathrm{v}\frac{\partial}{\partial z}\right)\rho_\mathrm{v}(z,t) = -i\bigg[H_\mathrm{v}(z, t), \rho_\mathrm{v}(z, t)\bigg] \end{equation} ~~~~~~ (1)\]

\(\rho_\mathrm{v}(z, t)\) is a complex valued \(2\times2\) matrix carrying the information about the number densities (diagonal entries) of the e-type (\(\nu_e\)) and the x-type (\(\nu_x\)) neutrinos, and the correlations (off-diagonal entries) among them for a given velocity mode \(v\) at position \(z\) and time \(t\). The density matrix \(\rho_\mathrm{v}\) takes the following form.

\[\rho_\mathrm{v}(z, t) = \left(\begin{align} &\rho_{ee} ~~ \rho_{ex} \\ &\rho_{ex}^* ~~ \rho_{xx} \end{align} \right). ~~~~~~ (2)\]

The quantity \(H_\mathrm{v}(z, t)\) on the right hand side of the equation (1) represents the Hamiltonian which dictates the dynamics of the flavor transitions. In general, \(H_\mathrm{v}\) contentains contributions from vacuum mixing \(H^{\text{vac}}\), interaction with matter \(H^{\text{m}}\) and the interactions among themselves \(H^{\nu\nu}\). In the present implementation of COSE\(\nu\), the contribution from matter has been neglected. Thus, \(H_\mathrm{v}\) takes the following form,

\[\begin{align} H_\mathrm{v}(z, t) &= H^{\text{vac}} + H^{\nu\nu} \\ &= H^{\text{vac}} + \mu\int_{-1}^{1}d\mathrm{v}'(1-\mathrm{v}\mathrm{v}')\left(\rho_\mathrm{v'}-\bar{\rho}_\mathrm{v'}^*\right), ~~~~~~ (3) \end{align}\]

where \(\bar\rho\) is the density matrix for antineutrinos.

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