Initialisation

This folder defines initialisation and equilibrium functions for the simulations.
For more details on each simulation, see simulations.

Diocotron instability

More details about the diocotron simulation are given in diocotron.

The initializatrion and equilibrium functions are defined in the DiocotronDensitySolution class.

Initialisation

We suppose as perturbed initial condition

  • \(\rho(0,r, \theta) = ( 1 + \epsilon \cos(l\theta)) \ \mathbb{I}_{ R_1 \leq r \leq R_2 }\)

where \(\mathbb{I}\) is the characteristic function.

In Edoardo Zoni's article [1], an exponential is also added to make the solution smoother:

  • \(\rho(0,r, \theta) = ( 1 + \epsilon \cos(l\theta)) \exp\left(- \left(\frac{r - \bar{r}}{d}\right)^p\right) \ \mathbb{I}_{ R_1 \leq r \leq R_2 }\)

with \(p = 50\), \(\bar{r} = \frac{R_1 + R_2}{2}\) and \(d = \frac{R_2 - R_1}{2}\). This last version is implemented in the code.

Equilibrium

The associated equilibrium is

  • \(\rho_0(r) = \mathbb{I}_{ R_1 \leq r \leq R_2 }\)

or

  • \(\rho_0(r) = \exp\left(- \left(\frac{r - \bar{r}}{d}\right)^p\right) \ \mathbb{I}_{ R_1 \leq r \leq R_2 }\) in Edoardo Zoni's article [1].

Dispersion relation

The DiocotronDensitySolution also solves the dispersion relation:

\[ \left(\frac{\omega}{\omega_D} \right)^2 - b_l \frac{\omega}{\omega_D} + c_l = 0 \]

with

\[ b_l \left( 1 - \frac{W_1^{2l}}{W_2^{2l}} \right) = l\left[ 1 - \frac{R_1^2}{R_2^2} + \frac{\omega_q }{\omega_D} \left( 1 + \frac{R_1^2}{R_2^2} \right)\right]\left[ 1 - \frac{W_1^{2l}}{W_2^{2l}} \right] + \left(1 - \frac{R_1^{2l}}{R_2^{2l}} \right) \left(\frac{R_2^{2l}}{W_2^{2l}} - \frac{W_1^{2l}}{R_1^{2l}} \right) \]
\[ c_l \left( 1 - \frac{W_1^{2l}}{W_2^{2l}} \right) = l^2 \frac{\omega_q }{\omega_D} \left[1 - \frac{R_1^2}{R_2^2} + \frac{\omega_q }{\omega_D} \frac{R_1^2}{R_2^2}\right]\left( 1 - \frac{W_1^{2l}}{W_2^{2l}} \right) -l \frac{\omega_q }{\omega_D} \left( 1 - \frac{W_1^{2l}}{R_2^{2l}} \right)\left( 1 - \frac{R_2^{2l}}{W_2^{2l}} \right) \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + l \left[1 - \frac{R_1^2}{R_2^2} + \frac{\omega_q }{\omega_D} \frac{R_1^2}{R_2^2} \right]\left( 1 - \frac{R_1^{2l}}{W_2^{2l}} \right) \left( 1 - \frac{W_1^{2l}}{R_1^{2l}} \right) - \left( 1 - \frac{R_2^{2l}}{W_2^{2l}} \right) \left( 1 - \frac{W_1^{2l}}{R_1^{2l}} \right) \left( 1 - \frac{R_1^{2l}}{R_2^{2l}} \right) \]

The imaginary part of the dispersion relation solution is given by DiocotronDensitySolution::get_slope() and its real part by DiocotronDensitySolution::get_frequency(). The imaginary part of the dispersion relation solution corresponds to the slope of the L2-norm perturbation in the linear phase (\(t \in [20, 50]\) s for the default parameters).

Vortex merger

More details about the diocotron simulation are given in vortex_merger.

Equilibrium

The equilibrium is determined by the eigenvalue problem of finding \((\sigma, \phi)\) such that

\[ - \nabla \cdot \nabla \phi = \sigma f(\phi), \]

with given \(f\) such that \(f'(\phi) \neq 0\). See Edoardo Zoni's article [1] for more details.

The algorithm for given initial data \((\sigma^0, \phi^0)\) is the following:

  • compute the density: \(\rho^{i} = \sigma^{i-1} f(\phi^{i-1})\),
  • compute a temporary electrical potential: \(- \nabla\cdot\nabla \phi_*^{i} = \rho^i\),
  • compute the coefficient:
  • if the maximum value \(\phi_{\text{max}}\) is given: \(c^i = \frac{\phi_{\text{max}}}{\Vert\phi_*^i\Vert_{\mathcal{L}^\infty}}\),
  • update the data: \((\sigma^i, \phi^i) = c^i (\sigma^{i-1}, \phi_*^i)\).

Then, we repeat all the steps until \(|\sigma^i - \sigma^{i-1}| \leq \tau\) for a given tolerance.

For the vortex merger simulation, we choose \(f(\phi)= \phi^2\) and \(\phi_{\text{max}} = 1\).

References

[1] Edoardo Zoni, Yaman Güçlü, "Solving hyperbolic-elliptic problems on singular mapped disk-like domains with the method of characteristics and spline finite elements", https://doi.org/10.1016/j.jcp.2019.108889, Journal of Computational Physics, 2019.

Contents

  • diocotron_initialisation_equilibrium.hpp : defines a DiocotronDensitySolution class which computes the initialisation and equilibrium functions.
  • vortex_merger_equilibrium.hpp : defines a VortexMergerEquilibria class which computes the equilibrium function.
  • vortex_merger_initialisation.hpp : defines a VortexMergerDensitySolution class which computes the initialisation function.