Models #

\[\hat v = c\,,\qquad \hat E = c \hat B \,.\]

\[\begin{split}\frac{1}{n_0} &\frac{\partial \mathbf j}{\partial t} = \frac{1}{\varepsilon} \mathbf E + \frac{1}{\varepsilon n_0} \mathbf j \times \mathbf B_0\,, \\[2mm] &\frac{\partial \mathbf B}{\partial t} + \nabla\times\mathbf E = 0\,, \\[2mm] -&\frac{\partial \mathbf E}{\partial t} + \nabla\times\mathbf B = \frac{\alpha^2}{\varepsilon} \mathbf j \,,\end{split}\]

where \((n_0,\mathbf B_0)\) denotes a (inhomogeneous) background and

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,, \qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t}\,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class Electrons[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

class struphy.models.ColdPlasmaVlasov[source]#

Bases: StruphyModel

Cold plasma hybrid model.

\[\hat v = c\,,\qquad \hat E = c \hat B \,,\qquad \hat f = \frac{\hat n}{c^3} \,.\]

\[\begin{split}&\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon_\textnormal{h}}\Big[ \mathbf{E} + \mathbf{v} \times \left( \mathbf{B} + \mathbf{B}_0 \right) \Big] \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,, \\[2mm] \frac{1}{n_0} &\frac{\partial \mathbf j_\textnormal{c}}{\partial t} = \frac{1}{\varepsilon_\textnormal{c}} \mathbf E + \frac{1}{\varepsilon_\textnormal{c} n_0} \mathbf j_\textnormal{c} \times \mathbf B_0\,, \\[2mm] &\frac{\partial \mathbf B}{\partial t} + \nabla\times\mathbf E = 0\,, \\[2mm] -&\frac{\partial \mathbf E}{\partial t} + \nabla\times\mathbf B = \frac{\alpha^2}{\varepsilon_\textnormal{h}} \left( \mathbf j_\textnormal{c} + \int_{\mathbb{R}^3} \mathbf{v} f \, \text{d}^3 \mathbf{v} \right) \,,\end{split}\]

where \((n_0,\mathbf B_0)\) denotes a (inhomogeneous) background and

\[\alpha = \frac{\hat \Omega_\textnormal{p,cold}}{\hat \Omega_\textnormal{c,cold}}\,, \qquad \varepsilon_\textnormal{c} = \frac{1}{\hat \Omega_\textnormal{c,cold} \hat t}\,, \qquad \varepsilon_\textnormal{h} = \frac{1}{\hat \Omega_\textnormal{c,hot} \hat t} \,.\]

At initial time the Poisson equation is solved once to weakly satisfy the Gauss law:

\[\begin{align} \nabla \cdot \mathbf{E} & = \nu \frac{\alpha^2}{\varepsilon_\textnormal{h}} \int_{\mathbb{R}^3} f \, \text{d}^3 \mathbf{v}\,. \end{align}\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class ThermalElectrons[source]#: Bases: FluidSpecies

class HotElectrons[source]#: Bases: ParticleSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.DeterministicParticleDiffusion[source]#

Bases: StruphyModel

Diffusion equation discretized with a deterministic particle method; the solution is \(L^2\)-projected onto \(V^0 \subset H^1\) to compute the flux.

\[\hat D := \frac{\hat x^2}{\hat t } \,.\]

Equations: Find \(u:\mathbb R\times \Omega\to \mathbb R^+\) such that

\[\frac{\partial u}{\partial t} + \nabla \cdot\left(\mathbf F(u) u\right) = 0\,, \qquad \mathbf F(u) = -\mathbb D\,\frac{\nabla u}{u}\,,\]

where \(\mathbb D: \Omega\to \mathbb R^{3\times 3 }\) is a positive diffusion matrix. At the moment only matrices of the form \(D*Id\) are implemented, where \(D > 0\) is a positive diffusion coefficient.

Propagators (called in sequence):

PushDeterministicDiffusion

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class Hydrogen[source]#: Bases: ParticleSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

class struphy.models.DriftKineticElectrostaticAdiabatic[source]#

Bases: StruphyModel

Drift-kinetic equation for one ion species in static background magnetic field, coupled to quasi-neutrality equation with adiabatic electrons.

\[\hat v = \hat v_\textrm{i} = \sqrt{\frac{k_B \hat T_\textrm{i}}{m_\textrm{i}}}\,,\qquad \hat E = \hat v_\textrm{i}\hat B\,,\qquad \hat \phi = \hat E \hat x \,.\]

\[\begin{split}&\frac{\partial f}{\partial t} + \left[ v_\parallel \frac{\mathbf{B}^*}{B^*_\parallel} + \frac{\mathbf{E}^* \times \mathbf{b}_0}{B^*_\parallel}\right] \cdot \frac{\partial f}{\partial \mathbf{X}} + \left[\frac{1}{\varepsilon} \frac{\mathbf{B}^*}{B^*_\parallel} \cdot \mathbf{E}^*\right] \cdot \frac{\partial f}{\partial v_\parallel} = 0\,. \\[2mm] - &\nabla_\perp \cdot \left( \frac{n_0}{|B_0|^2} \nabla_\perp \phi \right) + \frac{1}{\varepsilon} n_0 \left(1 + \frac{1}{Z \varepsilon} \frac{1}{T_{0}} \phi \right) = \frac 1 \varepsilon \int f B^*_\parallel \,\textnormal d v_\parallel \textnormal d \mu \,.\end{split}\]

where \(f(\mathbf{X}, v_\parallel, \mu, t)\) is the guiding center distribution and

\[\mathbf{E}^* = - \nabla \phi - \varepsilon \mu \nabla |B_0| \,, \qquad \mathbf{B}^* = \mathbf{B}_0 + \varepsilon v_\parallel \nabla \times \mathbf{b}_0 \,,\qquad B^*_\parallel = \mathbf B^* \cdot \mathbf b_0 \,,\]

and with the normalization parameters

\[\varepsilon := \frac{1}{\hat \Omega_\textrm{c} \hat t}\,,\qquad \hat \Omega_\textrm{c} = \frac{q_\textrm{i} \hat B}{m_\textrm{i}} \,.\]

Notes

The Control variate method in the Poisson equation is optional; in case it is enabled via the parameter file, the following Poisson equation is solved:

Find \(\phi \in H^1\) such that

\[\int \frac{n_0}{|B_0|^2} \nabla_\perp \psi \cdot \nabla_\perp \phi\,\textrm d \mathbf x + \frac{1}{Z\varepsilon^2} \int \frac{n_0}{T_{0}} \psi \phi \,\textrm d \mathbf x = \frac 1 \varepsilon \int \int \psi \, (f - f_0) B^*_\parallel \,\textrm d \mathbf x\,\textnormal d v_\parallel \textnormal d \mu \qquad \forall \ \psi \in H^1\,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class KineticIons[source]#: Bases: ParticleSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.EulerSPH(with_B0: bool = True)[source]#

Bases: StruphyModel

Euler equations discretized with smoothed particle hydrodynamics (SPH).

\[\hat u = \hat v_\textnormal{th} \,.\]

\[\begin{split}\begin{align} \partial_t \rho + \nabla \cdot (\rho \mathbf u) &= 0\,, \\[2mm] \rho(\partial_t \mathbf u + \mathbf u \cdot \nabla \mathbf u) &= - \nabla \left(\rho^2 \frac{\partial \mathcal U(\rho, S)}{\partial \rho} \right)\,, \\[2mm] \partial_t S + \mathbf u \cdot \nabla S &= 0\,, \end{align}\end{split}\]

where \(S\) denotes the entropy per unit mass. The internal energy per unit mass can be defined in two ways:

\[ \begin{align}\begin{aligned}\mathrm{"isothermal:"}\qquad &\mathcal U(\rho, S) = \kappa(S) \log \rho\,.\\\mathrm{"polytropic:"}\qquad &\mathcal U(\rho, S) = \kappa(S) \frac{\rho^{\gamma - 1}}{\gamma - 1}\,.\end{aligned}\end{align} \]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EulerFluid[source]#: Bases: ParticleSpecies

class Propagators(with_B0: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.GuidingCenter[source]#

Bases: StruphyModel

Guiding-center equation in static background magnetic field.

\[\hat v = \hat v_\textnormal{A} \,.\]

\[\frac{\partial f}{\partial t} + \left[ v_\parallel \frac{\mathbf{B}^*}{B^*_\parallel} + \frac{\mathbf{E}^* \times \mathbf{b}_0}{B^*_\parallel}\right] \cdot \frac{\partial f}{\partial \mathbf{X}} + \left[\frac{1}{\epsilon} \frac{\mathbf{B}^*}{B^*_\parallel} \cdot \mathbf{E}^*\right] \cdot \frac{\partial f}{\partial v_\parallel} = 0\,.\]

where \(f(\mathbf{X}, v_\parallel, \mu, t)\) is the guiding center distribution and

\[\mathbf{E}^* = -\epsilon \mu \nabla |B_0| \,, \qquad \mathbf{B}^* = \mathbf{B}_0 + \epsilon v_\parallel \nabla \times \mathbf{b}_0 \,,\qquad B^*_\parallel = \mathbf B^* \cdot \mathbf b_0 \,.\]

Moreover,

\[\epsilon = \frac{1 }{ \hat \Omega_{\textnormal{c}} \hat t}\,,\qquad \textnormal{with} \qquad\hat \Omega_{\textnormal{c}} = \frac{Ze \hat B}{A m_\textnormal{H}}\,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class KineticIons[source]#: Bases: ParticleSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

class struphy.models.HasegawaWakatani[source]#

Bases: StruphyModel

Hasegawa-Wakatani equations in 2D.

\[\hat u = \hat v_\textnormal{th}\,,\qquad \hat \phi = \hat u\, \hat x \,.\]

\[\begin{split}&\frac{\partial n}{\partial t} = C (\phi - n) - [\phi, n] - \kappa\, \partial_y \phi + \nu\, \nabla^{2N} n\,, \\[2mm] &\frac{\partial \omega}{\partial t} = C (\phi - n) - [\phi, \omega] + \nu\, \nabla^{2N} \omega \,, \\[3mm] &\Delta \phi = \omega\,,\end{split}\]

where \([\phi, n] = \partial_x \phi \partial_y n - \partial_y \phi \partial_x n\), \(C = C(x, y)\) and \(\kappa\) and \(\nu\) are constants (at the moment only \(N=1\) is available).

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class Plasma[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

update_rho()[source]#

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.LinearExtendedMHDuniform[source]#

Bases: StruphyModel

Linear extended MHD with zero-flow equilibrium (\(\mathbf U_0 = 0\)). For uniform background conditions only.

\[\hat U = \hat v_\textnormal{A} \,.\]

\[\begin{split}&\frac{\partial \tilde \rho}{\partial t}+\nabla\cdot(\rho_0 \tilde{\mathbf{U}})=0\,, \\[2mm] \rho_0&\frac{\partial \tilde{\mathbf{U}}}{\partial t} + \nabla \tilde p =(\nabla\times \tilde{\mathbf{B}})\times\mathbf{B}_0 \,, \\[2mm] &\frac{\partial \tilde p}{\partial t} + \frac{5}{3}\,p_{0}\nabla\cdot \tilde{\mathbf{U}}=0\,, \\[2mm] &\frac{\partial \tilde{\mathbf{B}}}{\partial t} - \nabla\times \left( \tilde{\mathbf{U}} \times \mathbf{B}_0 - \frac{1}{\varepsilon} \frac{\nabla\times \tilde{\mathbf{B}}}{\rho_0}\times \mathbf{B}_0 \right) = 0\,.\end{split}\]

where

\[\varepsilon = \frac{1}{\hat \Omega_{\textnormal{c}} \hat t}\,,\qquad \textnormal{with} \qquad\hat \Omega_{\textnormal{c}} = \frac{Ze \hat B}{A m_\textnormal{H}}\,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.LinearMHD[source]#

Bases: StruphyModel

Linear ideal MHD with zero-flow equilibrium (\(\mathbf U_0 = 0\)).

\[\hat U = \hat v_\textnormal{A} \,.\]

\[\begin{split}&\frac{\partial \tilde \rho}{\partial t}+\nabla\cdot(\rho_0 \tilde{\mathbf{U}})=0\,, \\[2mm] \rho_0&\frac{\partial \tilde{\mathbf{U}}}{\partial t} + \nabla \tilde p = (\nabla \times \tilde{\mathbf{B}})\times \mathbf{B}_0 + (\nabla\times\mathbf{B}_0)\times \tilde{\mathbf{B}} \,, \\[2mm] &\frac{\partial \tilde p}{\partial t} + \nabla\cdot(p_0 \tilde{\mathbf{U}}) + \frac{2}{3}\,p_0\nabla\cdot \tilde{\mathbf{U}}=0\,, \\[2mm] &\frac{\partial \tilde{\mathbf{B}}}{\partial t} - \nabla\times(\tilde{\mathbf{U}} \times \mathbf{B}_0) = 0\,.\end{split}\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.LinearMHDDriftkineticCC(turn_off: tuple[str, ...] = (None,))[source]#

Bases: StruphyModel

Hybrid linear ideal MHD + energetic ions (5D Driftkinetic) with current coupling scheme.

\[\hat U = \hat v =: \hat v_\textnormal{A, bulk} \,, \qquad \hat f_\textnormal{h} = \frac{\hat n}{\hat v_\textnormal{h} \hat \mu \hat B} \,,\qquad \hat \mu = \frac{A_\textnormal{h} m_\textnormal{H} \hat v_\textnormal{h}^2}{\hat B} \,.\]

\[\begin{split}\begin{align} \textnormal{MHD} &\left\{ \begin{aligned} &\frac{\partial \tilde{\rho}}{\partial t}+\nabla\cdot(\rho_{0} \tilde{\mathbf{U}})=0\,, \\ \rho_{0} &\frac{\partial \tilde{\mathbf{U}}}{\partial t} - \tilde p\, \nabla = (\nabla \times \tilde{\mathbf{B}}) \times \mathbf{B} + (\nabla \times \mathbf B_0) \times \tilde{\mathbf{B}} + \frac{A_\textnormal{h}}{A_\textnormal{b}} \left[ \frac{1}{\epsilon} n_\textnormal{gc} \tilde{\mathbf{U}} - \frac{1}{\epsilon} \mathbf{J}_\textnormal{gc} - \nabla \times \mathbf{M}_\textnormal{gc} \right] \times \mathbf{B} \,, \\ &\frac{\partial \tilde p}{\partial t} + \nabla\cdot(p_0 \tilde{\mathbf{U}}) + \frac{2}{3}\,p_0\nabla\cdot \tilde{\mathbf{U}}=0\,, \\ &\frac{\partial \tilde{\mathbf{B}}}{\partial t} - \nabla\times(\tilde{\mathbf{U}} \times \mathbf{B}) = 0\,, \end{aligned} \right. \\[2mm] \textnormal{EPs}\,\, &\left\{\,\, \begin{aligned} \quad &\frac{\partial f_\textnormal{h}}{\partial t} + \frac{1}{B_\parallel^*}(v_\parallel \mathbf{B}^* - \mathbf{b}_0 \times \mathbf{E}^*)\cdot\nabla f_\textnormal{h} + \frac{1}{\epsilon} \frac{1}{B_\parallel^*} (\mathbf{B}^* \cdot \mathbf{E}^*) \frac{\partial f_\textnormal{h}}{\partial v_\parallel} = 0\,, \\ & n_\textnormal{gc} = \int f_\textnormal{h} B_\parallel^* \,\textnormal dv_\parallel \textnormal d\mu \,, \\ & \mathbf{J}_\textnormal{gc} = \int \frac{f_\textnormal{h}}{B_\parallel^*}(v_\parallel \mathbf{B}^* - \mathbf{b}_0 \times \mathbf{E}^*) \,\textnormal dv_\parallel \textnormal d\mu \,, \\ & \mathbf{M}_\textnormal{gc} = - \int f_\textnormal{h} B_\parallel^* \mu \mathbf{b}_0 \,\textnormal dv_\parallel \textnormal d\mu \,, \end{aligned} \right. \end{align}\end{split}\]

where

\[\begin{split}\begin{align} B^*_\parallel = \mathbf{b}_0 \cdot \mathbf{B}^*\,, \\[2mm] \mathbf{B}^* &= \mathbf{B} + \epsilon v_\parallel \nabla \times \mathbf{b}_0 \,, \\[2mm] \mathbf{E}^* &= - \tilde{\mathbf{U}} \times \mathbf{B} - \epsilon \mu \nabla (\mathbf{b}_0 \cdot \mathbf{B}) \,, \end{align}\end{split}\]

with the normalization parameter

\[\epsilon = \frac{1}{\hat \Omega_\textnormal{c,hot} \hat t} \,, \qquad \hat \Omega_\textnormal{c,hot} = \frac{Z_\textnormal{h} e \hat B}{A_\textnormal{h} m_\textnormal{H}} \,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EnergeticIons[source]#: Bases: ParticleSpecies

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Propagators(turn_off: tuple[str, ...] = (None,))[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.LinearMHDVlasovCC[source]#

Bases: StruphyModel

Hybrid linear MHD + energetic ions (6D Vlasov) with current coupling scheme.

\[\hat U = \hat v = \hat v_\textnormal{A} \,, \qquad \hat f_\textnormal{h} = \frac{\hat n}{\hat v_\textnormal{A}^3} \,.\]

\[\begin{split}\begin{align} \textnormal{MHD}\,\, &\left\{\,\, \begin{aligned} &\frac{\partial \tilde{\rho}}{\partial t}+\nabla\cdot(\rho_0 \tilde{\mathbf{U}})=0\,, \\[2mm] \rho_0 &\frac{\partial \tilde{\mathbf{U}}}{\partial t} + \nabla \tilde p =(\nabla\times \tilde{\mathbf{B}})\times\mathbf{B}_0 + \mathbf{J}_0\times \tilde{\mathbf{B}} \color{blue} + \frac{A_\textnormal{h}}{A_\textnormal{b}} \frac{1}{\varepsilon} \left(n_\textnormal{h}\tilde{\mathbf{U}}-n_\textnormal{h}\mathbf{u}_\textnormal{h}\right)\times(\mathbf{B}_0+\tilde{\mathbf{B}}) \color{black}\,, \\[2mm] &\frac{\partial \tilde p}{\partial t} + (\gamma-1)\nabla\cdot(p_0 \tilde{\mathbf{U}}) + p_0\nabla\cdot \tilde{\mathbf{U}}=0\,, \\[2mm] &\frac{\partial \tilde{\mathbf{B}}}{\partial t} = \nabla\times(\tilde{\mathbf{U}} \times \mathbf{B}_0)\,,\qquad \nabla\cdot\tilde{\mathbf{B}}=0\,, \end{aligned} \right. \\[2mm] \textnormal{EPs}\,\, &\left\{\,\, \begin{aligned} &\quad\,\,\frac{\partial f_\textnormal{h}}{\partial t}+\mathbf{v}\cdot\nabla f_\textnormal{h} + \frac{1}{\varepsilon} \left[\color{blue} (\mathbf{B}_0+\tilde{\mathbf{B}})\times\tilde{\mathbf{U}} \color{black} + \mathbf{v}\times(\mathbf{B}_0+\tilde{\mathbf{B}})\right]\cdot \frac{\partial f_\textnormal{h}}{\partial \mathbf{v}} =0\,, \\[2mm] &\quad\,\,n_\textnormal{h}=\int_{\mathbb{R}^3}f_\textnormal{h}\,\textnormal{d}^3 \mathbf v\,,\qquad n_\textnormal{h}\mathbf{u}_\textnormal{h}=\int_{\mathbb{R}^3}f_\textnormal{h}\mathbf{v}\,\textnormal{d}^3 \mathbf v\,, \end{aligned} \right. \end{align}\end{split}\]

where \(\mathbf{J}_0 = \nabla\times\mathbf{B}_0\) and

\[\varepsilon = \frac{1}{\hat \Omega_{\textnormal{c,hot}} \hat t}\,,\qquad \textnormal{with} \qquad\hat \Omega_{\textnormal{c,hot}} = \frac{Z_\textnormal{h}e \hat B}{A_\textnormal{h} m_\textnormal{H}}\,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class EnergeticIons[source]#: Bases: ParticleSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.LinearMHDVlasovPC(turn_off: tuple[str, ...] = (None,))[source]#

Bases: StruphyModel

Hybrid linear MHD + energetic ions (6D Vlasov) with pressure coupling scheme.

\[\hat U = \hat v =: \hat v_\textnormal{A, bulk} \,, \qquad \hat f_\textnormal{h} = \frac{\hat n}{\hat v_\textnormal{A}^3} \,,\qquad \hat{\mathbb{P}}_\textnormal{h} = A_\textnormal{h}m_\textnormal{H}\hat n \hat v_\textnormal{A}^2\,,\]

Implemented equations:

\[\begin{split}\begin{align} \textnormal{MHD} &\left\{ \begin{aligned} &\frac{\partial \tilde{\rho}}{\partial t}+\nabla\cdot(\rho_0 \tilde{\mathbf{U}})=0\,, \\ \rho_0 &\frac{\partial \tilde{\mathbf{U}}}{\partial t} + \nabla \tilde p + \frac{A_\textnormal{h}}{A_\textnormal{b}} \nabla\cdot \tilde{\mathbb{P}}_{\textnormal{h},\perp} =(\nabla\times \tilde{\mathbf{B}})\times\mathbf{B}_0 + \mathbf{J}_0\times \tilde{\mathbf{B}} \,, \qquad \mathbf{J}_0 = \nabla\times\mathbf{B}_0\,, \\ &\frac{\partial \tilde p}{\partial t} + \nabla\cdot(p_0 \tilde{\mathbf{U}}) + \frac{2}{3}\,p_0\nabla\cdot \tilde{\mathbf{U}}=0\,, \\ &\frac{\partial \tilde{\mathbf{B}}}{\partial t} - \nabla\times(\tilde{\mathbf{U}} \times \mathbf{B}_0) = 0\,, \end{aligned} \right. \\[2mm] \textnormal{EPs}\,\, &\left\{\,\, \begin{aligned} &\quad\,\,\frac{\partial f_\textnormal{h}}{\partial t} + (\mathbf{v} + \tilde{\mathbf{U}}_\perp)\cdot \nabla f_\textnormal{h} + \left[\frac{1}{\epsilon}\, \mathbf{v}\times(\mathbf{B}_0 + \tilde{\mathbf{B}}) - \nabla \tilde{\mathbf{U}}_\perp\cdot \mathbf{v} \right]\cdot \frac{\partial f_\textnormal{h}}{\partial \mathbf{v}} = 0\,, \\ &\quad\,\,\tilde{\mathbb{P}}_{\textnormal{h},\perp} = \int \mathbf{v}_\perp\mathbf{v}^\top_\perp f_\textnormal{h} d\mathbf{v} \,, \end{aligned} \right. \end{align}\end{split}\]

where

\[\epsilon = \frac{\hat \omega}{2 \pi \, \hat \Omega_{\textnormal{c,hot}}} \,,\qquad \textnormal{with} \qquad\hat \Omega_{\textnormal{c,hot}} = \frac{Z_\textnormal{h}e \hat B}{A_\textnormal{h} m_\textnormal{H}}\,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EnergeticIons[source]#: Bases: ParticleSpecies

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Propagators(turn_off: tuple[str, ...] = (None,))[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.LinearVlasovAmpereOneSpecies(with_B0: bool = True, with_E0: bool = True)[source]#

Bases: StruphyModel

Linearized Vlasov-Ampère equations for one species.

\[\begin{align} \hat v = c \,, \qquad \hat E = \hat B \hat v\,,\qquad \hat \phi = \hat E \hat x \,. \end{align}\]

\[\begin{split}\begin{align} & \frac{\partial \tilde{\mathbf E}}{\partial t} = - \frac{\alpha^2}{\varepsilon} \int_{\mathbb R^3} \mathbf{v} \tilde f\, \textrm d^3 \mathbf v \,, \\[2mm] & \frac{\partial \tilde f}{\partial t} + \mathbf{v} \cdot \, \nabla \tilde f + \frac{1}{\varepsilon} \left( \mathbf{E}_0 + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial \tilde f}{\partial \mathbf{v}} = \frac{1}{v_{\text{th}}^2 \varepsilon} \, \tilde{\mathbf E} \cdot \mathbf{v} f_0 \,, \end{align}\end{split}\]

with the normalization parameter

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,,\qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \qquad \hat\Omega_\textnormal{p} = \sqrt{\frac{\hat n (Ze)^2}{\epsilon_0 (A m_\textnormal{H})}} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

where \(Z=-1\) and \(A=1/1836\) for electrons. The background distribution function \(f_0\) is a uniform Maxwellian

\[f_0 = \frac{n_0(\mathbf{x})}{\left( \sqrt{2 \pi} v_{\text{th}} \right)^3} \exp \left( - \frac{|\mathbf{v}|^2}{2 v_{\text{th}}^2} \right) \,,\]

and the background electric field has to verify the following compatibility condition between with background density

\[\nabla_{\mathbf{x}} \ln (n_0(\mathbf{x})) = \frac{1}{v_{\text{th}}^2 \varepsilon} \mathbf{E}_0 \,.\]

At initial time the weak Poisson equation is solved once to weakly satisfy Gauss’ law,

\[\begin{split}\begin{align} \int_\Omega \nabla \psi^\top \cdot \nabla \phi \,\textrm d \mathbf x &= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi\, \tilde f \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,, \\[2mm] \tilde{\mathbf{E}}(t=0) &= -\nabla \phi(t=0) \,. \end{align}\end{split}\]

Moreover, it is assumed that

\[\int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v} = 0 \,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class KineticIons[source]#: Bases: ParticleSpecies

class Propagators(with_B0: bool = True, with_E0: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.LinearVlasovMaxwellOneSpecies(with_B0: bool = True, with_E0: bool = True)[source]#

Bases: LinearVlasovAmpereOneSpecies

Linearized Vlasov-Ampère equations for one species.

\[\begin{align} \hat v = c \,, \qquad \hat E = \hat B \hat v\,,\qquad \hat \phi = \hat E \hat x \,. \end{align}\]

\[\begin{split}\begin{align} & \frac{\partial \tilde{\mathbf E}}{\partial t} = \nabla \times \tilde{\mathbf B} - \frac{\alpha^2}{\varepsilon} \int_{\mathbb R^3}\mathbf{v} \tilde f\, \textrm d^3 \mathbf v \,, \\[2mm] & \frac{\partial \tilde{\mathbf B}}{\partial t} = - \nabla \times \tilde{\mathbf E} \,, \\[2mm] & \frac{\partial \tilde f}{\partial t} + \mathbf{v} \cdot \, \nabla \tilde f + \frac{1}{\varepsilon} \left( \mathbf{E}_0 + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial \tilde f}{\partial \mathbf{v}} = \frac{1}{v_{\text{th}}^2 \varepsilon} \, \tilde{\mathbf E} \cdot \mathbf{v} f_0 \,, \end{align}\end{split}\]

with the normalization parameter

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,,\qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \qquad \hat\Omega_\textnormal{p} = \sqrt{\frac{\hat n (Ze)^2}{\epsilon_0 (A m_\textnormal{H})}} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

where \(Z=-1\) and \(A=1/1836\) for electrons. The background distribution function \(f_0\) is a uniform Maxwellian

\[f_0 = \frac{n_0(\mathbf{x})}{\left( \sqrt{2 \pi} v_{\text{th}} \right)^3} \exp \left( - \frac{|\mathbf{v}|^2}{2 v_{\text{th}}^2} \right) \,,\]

and the background electric field has to verify the following compatibility condition between with background density

\[\nabla_{\mathbf{x}} \ln (n_0(\mathbf{x})) = \frac{1}{v_{\text{th}}^2 \varepsilon} \mathbf{E}_0 \,.\]

At initial time the weak Poisson equation is solved once to weakly satisfy Gauss’ law,

\[\begin{split}\begin{align} \int_\Omega \nabla \psi^\top \cdot \nabla \phi \,\textrm d \mathbf x &= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi\, \tilde f \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,, \\[2mm] \tilde{\mathbf{E}(t=0)} &= -\nabla \phi(t=0) \,. \end{align}\end{split}\]

Moreover, it is assumed that

\[\int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v} = 0 \,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class KineticIons[source]#: Bases: ParticleSpecies

class Propagators(with_B0: bool = True, with_E0: bool = True)[source]#: Bases: object

class struphy.models.Maxwell[source]#

Bases: StruphyModel

Maxwell’s equations in vacuum.

\[\hat E = c \hat B\,.\]

\[ \begin{align}\begin{aligned}&\frac{\partial \mathbf E}{\partial t} - \nabla\times\mathbf B = 0\,,\\&\frac{\partial \mathbf B}{\partial t} + \nabla\times\mathbf E = 0\,.\end{aligned}\end{align} \]

Propagators (called in sequence):

Maxwell

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

class struphy.models.Poisson[source]#

Bases: StruphyModel

Weak discretization of Poisson’s equation with diffusion matrix, stabilization and time-depedent right-hand side.

\[\hat D = \frac{\hat n}{\hat x^2}\,,\qquad \hat \rho = \hat n \,.\]

Equations: Find \(\phi \in H^1\) such that

\[- \nabla \cdot D_0(\mathbf x) \nabla \phi + n_0(\mathbf x) \phi = \rho(t, \mathbf x)\,,\]

where \(n_0, \rho(t):\Omega \to \mathbb R\) are real-valued functions, \(\rho(t)\) parametrized with time \(t\), and \(D_0:\Omega \to \mathbb R^{3\times 3}\) is a positive matrix. Boundary terms from integration by parts are assumed to vanish.

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.PressureLessSPH[source]#

Bases: StruphyModel

Pressureless fluid discretized with smoothed particle hydrodynamics

\[\begin{split}&\partial_t \rho + \nabla \cdot ( \rho \mathbf u ) = 0 \,, \\[4mm] &\partial_t (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u) = - \nabla \phi_0 \,,\end{split}\]

where \(\phi_0\) is a static external potential.

Propagators (called in sequence):

PushEta

This is discretized by particles going in straight lines.

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class ColdFluid[source]#: Bases: ParticleSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.RandomParticleDiffusion[source]#

Bases: StruphyModel

Diffusion equation discretized with a (random) particle method; the diffusion is computed through a Wiener process.

\[\hat D := \frac{\hat x^2}{\hat t } \,.\]

Equations: Find \(u:\mathbb R\times \Omega\to \mathbb R^+\) such that

\[\frac{\partial u}{\partial t} - D \, \Delta u = 0\,,\]

where \(D > 0\) is a positive diffusion coefficient.

Propagators (called in sequence):

PushRandomDiffusion

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class Hydrogen[source]#: Bases: ParticleSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

class struphy.models.ShearAlfven[source]#

Bases: StruphyModel

ShearAlfven propagator from LinearMHD with zero-flow equilibrium (\(\mathbf U_0 = 0\)).

\[\hat U = \hat v_\textnormal{A} \,.\]

\[ \begin{align}\begin{aligned}\rho_0&\frac{\partial \tilde{\mathbf{U}}}{\partial t} =(\nabla\times \tilde{\mathbf{B}})\times\mathbf{B}_0\,,\\&\frac{\partial \tilde{\mathbf{B}}}{\partial t} - \nabla\times(\tilde{\mathbf{U}} \times \mathbf{B}_0) = 0\,.\end{aligned}\end{align} \]

Propagators (called in sequence):

ShearAlfven

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

class struphy.models.TwoFluidQuasiNeutralToy[source]#

Bases: StruphyModel

Linearized, quasi-neutral two-fluid model with zero electron inertia.

\[\hat u = \hat v_\textnormal{th}\,,\qquad e\hat \phi = m \hat v_\textnormal{th}^2\,.\]

\[\begin{split}\frac{\partial \mathbf u}{\partial t} &= - \nabla \phi + \frac{\mathbf u \times \mathbf B_0}{\varepsilon} + \nu \Delta \mathbf u + \mathbf f\,, \\[2mm] 0 &= \nabla \phi - \frac{\mathbf u_e \times \mathbf B_0}{\varepsilon} + \nu_e \Delta \mathbf u_e + \mathbf f_e \,, \\[3mm] \nabla & \cdot (\mathbf u - \mathbf u_e) = 0\,,\end{split}\]

where \(\mathbf B_0\) is a static magnetic field and \(\mathbf f, \mathbf f_e\) are given forcing terms, and with the normalization parameter

\[\varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

Propagators (called in sequence):

TwoFluidQuasiNeutralFull

References

[1] Juan Vicente Gutiérrez-Santacreu, Omar Maj, Marco Restelli: Finite element discretization of a Stokes-like model arising in plasma physics, Journal of Computational Physics 2018.

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMfields[source]#: Bases: FieldSpecies

class Ions[source]#: Bases: FluidSpecies

class Electrons[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.VariationalBarotropicFluid[source]#

Bases: StruphyModel

Barotropic fluid equations discretized with a variational method.

\[\hat u = \hat v_\textnormal{A} \qquad \hat{\mathcal U} = \frac{\hat \rho}{2} \,.\]

\[\begin{split}&\partial_t \rho + \nabla \cdot ( \rho \mathbf u ) = 0 \,, \\[4mm] &\partial_t (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u) + \rho \nabla \frac{(\rho \mathcal U (\rho))}{\partial \rho} = 0 \,.\end{split}\]

where the internal energy per unit mass is \(\mathcal U(\rho) = \rho/2\).

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class Fluid[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.VariationalCompressibleFluid[source]#

Bases: StruphyModel

Fully compressible fluid equations discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,, \qquad \hat{\mathcal U} = K\,,\qquad \hat s = \hat \rho C_v \,.\]

\[\begin{split}&\partial_t \rho + \nabla \cdot ( \rho \mathbf u ) = 0 \,, \\[4mm] &\partial_t (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u) + \rho \nabla \frac{(\rho \mathcal U (\rho, s))}{\partial \rho} + s \nabla \frac{(\rho \mathcal U (\rho, s))}{\partial s} = 0 \,, \\[4mm] &\partial_t s + \nabla \cdot ( s \mathbf u ) = 0 \,,\end{split}\]

where the internal energy per unit mass is \(\mathcal U(\rho) = \rho^{\gamma-1} \exp(s / \rho)\).

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class Fluid[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.VariationalPressurelessFluid[source]#

Bases: StruphyModel

Pressure-less fluid equations discretized with a variational method.

\[\hat u = \hat v_\textnormal{A} \,.\]

\[\begin{split}&\partial_t \rho + \nabla \cdot ( \rho \mathbf u ) = 0 \,, \\[4mm] &\partial_t (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u) = 0 \,.\end{split}\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class Fluid[source]#: Bases: FluidSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscoResistiveDeltafMHD(with_viscosity: bool = True, with_resistivity: bool = True)[source]#

Bases: StruphyModel

\(\delta f\) visco-resistive MHD equations discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,.\]

\[\begin{split}&\partial_t \tilde{\rho} + \nabla \cdot ( (\tilde{\rho}+\rho_0) \tilde{\mathbf u} ) = 0 \,, \\[4mm] &\partial_t ((\tilde{\rho}+\rho_0) \tilde{\mathbf u}) + \nabla \cdot ((\tilde{\rho}+\rho_0) \tilde{\mathbf u} \otimes \tilde{\mathbf u}) + \frac{1}{\gamma -1} \nabla \tilde{p} + \mathbf B_0 \times \nabla \times \tilde{\mathbf B} + \tilde{\mathbf B} \times \nabla \times \mathbf B_0 + \tilde{\mathbf B} \times \nabla \times \tilde{\mathbf B} - \nabla \cdot \left((\mu+\mu_a(\mathbf x)) \nabla \tilde{\mathbf u} \right) = 0 \,, \\[4mm] &\partial_t \tilde{p} + \tilde{\mathbf u} \cdot \nabla (\tilde{p} + p_0) + \gamma (\tilde{p} + p_0) \nabla \cdot \tilde{\mathbf u} = \frac{1}{(\gamma -1)}\left((\mu+\mu_a(\mathbf x)) |\nabla \tilde{\mathbf u}|^2 + (\eta + \eta_a(\mathbf x)) |\nabla \times \tilde{\mathbf B}|^2\right) \,, \\[4mm] &\partial_t \tilde{\mathbf B} + \nabla \times ( (\tilde{\mathbf B} + \mathbf B_0) \times \tilde{\mathbf u} ) + \nabla \times (\eta + \eta_a(\mathbf x)) \nabla \times \tilde{\mathbf B} = 0 \,,\end{split}\]

and \(\mu_a(\mathbf x)\) and \(\eta_a(\mathbf x)\) are artificial viscosity and resistivity coefficients.

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Diagnostics[source]#: Bases: DiagnosticSpecies

class Propagators(with_viscosity: bool = True, with_resistivity: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscoResistiveDeltafMHD_with_q(with_viscosity: bool = True, with_resistivity: bool = True)[source]#

Bases: StruphyModel

Linear visco-resistive MHD equations discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,.\]

\[\begin{split}&\partial_t \tilde{\rho} + \nabla \cdot ( \rho_0 \tilde{\mathbf u} ) = 0 \,, \\[4mm] &\partial_t (\rho_0 \tilde{\mathbf u}) + \frac{2 q_0}{\gamma -1} \nabla \tilde{q} + \frac{2 \tilde{q}}{\gamma -1} \nabla q_0 + \frac{2 \tilde{q}}{\gamma -1} \nabla \tilde{q} + \mathbf B_0 \times \nabla \times \tilde{\mathbf B} + \tilde{\mathbf B} \times \nabla \times \mathbf B_0 - \nabla \cdot \left((\mu+\mu_a(\mathbf x)) \nabla \tilde{\mathbf u} \right) = 0 \,, \\[4mm] &\partial_t \tilde{q} + \cdot(\nabla (q_0 + \tilde{q}) \mathbf u) + (\gamma/2 -1) (q_0 + \tilde{q}) \nabla \cdot u = 0 \,, \\[4mm] &\partial_t \tilde{\mathbf B} + \nabla \times ( \mathbf (B_0 + \tilde{\mathbf B}) \times \tilde{\mathbf u} ) + \nabla \times (\eta + \eta_a(\mathbf x)) \nabla \times \tilde{\mathbf B} = 0 \,,\end{split}\]

and \(\mu_a(\mathbf x)\) and \(\eta_a(\mathbf x)\) are artificial viscosity and resistivity coefficients.

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Diagnostics[source]#: Bases: DiagnosticSpecies

class Propagators(with_viscosity: bool = True, with_resistivity: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscoResistiveLinearMHD(with_viscosity: bool = True, with_resistivity: bool = True)[source]#

Bases: StruphyModel

Linear visco-resistive MHD equations discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,.\]

\[\begin{split}&\partial_t \tilde{\rho} + \nabla \cdot ( \rho_0 \tilde{\mathbf u} ) = 0 \,, \\[4mm] &\partial_t (\rho_0 \tilde{\mathbf u}) + \frac{1}{\gamma -1} \nabla \tilde{p} + \mathbf B_0 \times \nabla \times \tilde{\mathbf B} + \tilde{\mathbf B} \times \nabla \times \mathbf B_0 - \nabla \cdot \left((\mu+\mu_a(\mathbf x)) \nabla \tilde{\mathbf u} \right) = 0 \,, \\[4mm] &\partial_t \tilde{p} + \tilde{\mathbf u} \cdot \nabla p_0 + \gamma p_0 \nabla \cdot \tilde{\mathbf u} = \frac{1}{(\gamma -1)}\left((\mu+\mu_a(\mathbf x)) |\nabla \tilde{\mathbf u}|^2 + (\eta + \eta_a(\mathbf x)) |\nabla \times \tilde{\mathbf B}|^2\right) \,, \\[4mm] &\partial_t \tilde{\mathbf B} + \nabla \times ( \mathbf B_0 \times \tilde{\mathbf u} ) + \nabla \times (\eta + \eta_a(\mathbf x)) \nabla \times \tilde{\mathbf B} = 0 \,,\end{split}\]

and \(\mu_a(\mathbf x)\) and \(\eta_a(\mathbf x)\) are artificial viscosity and resistivity coefficients.

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Diagnostics[source]#: Bases: DiagnosticSpecies

class Propagators(with_viscosity: bool = True, with_resistivity: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscoResistiveLinearMHD_with_q(with_viscosity: bool = True, with_resistivity: bool = True)[source]#

Bases: StruphyModel

Linear visco-resistive MHD equations, with the q variable (square root of the pressure), discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,.\]

\[\begin{split}&\partial_t \tilde{\rho} + \nabla \cdot ( \rho_0 \tilde{\mathbf u} ) = 0 \,, \\[4mm] &\partial_t (\rho_0 \tilde{\mathbf u}) + \frac{2 q_0}{\gamma -1} \nabla \tilde{q} + \frac{2 \tilde{q}}{\gamma -1} \nabla q_0 + \mathbf B_0 \times \nabla \times \tilde{\mathbf B} + \tilde{\mathbf B} \times \nabla \times \mathbf B_0 - \nabla \cdot \left((\mu+\mu_a(\mathbf x)) \nabla \tilde{\mathbf u} \right) = 0 \,, \\[4mm] &\partial_t \tilde{q} + \cdot(\nabla q_0 \mathbf u) + (\gamma/2 -1) q_0 \nabla \cdot u = 0 \,, \\[4mm] &\partial_t \tilde{\mathbf B} + \nabla \times ( \mathbf B_0 \times \tilde{\mathbf u} ) + \nabla \times (\eta + \eta_a(\mathbf x)) \nabla \times \tilde{\mathbf B} = 0 \,,\end{split}\]

and \(\mu_a(\mathbf x)\) and \(\eta_a(\mathbf x)\) are artificial viscosity and resistivity coefficients.

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Diagnostics[source]#: Bases: DiagnosticSpecies

class Propagators(with_viscosity: bool = True, with_resistivity: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscoResistiveMHD(with_viscosity: bool = True, with_resistivity: bool = True)[source]#

Bases: StruphyModel

Full (non-linear) visco-resistive MHD equations discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,, \qquad \hat{\mathcal U} = \frac{\hat{\mathbf B}^2}{\hat \rho \mu_0 (\gamma-1)} \,,\qquad \hat s = \hat \rho\ \textrm{ln}\left(\frac{\hat{\mathbf B}^2}{\mu_0 (\gamma -1) \hat{\rho}}\right) \,.\]

\[\begin{split}&\partial_t \rho + \nabla \cdot ( \rho \mathbf u ) = 0 \,, \\[4mm] &\partial_t (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u) + \rho \nabla \frac{(\rho \mathcal U (\rho, s))}{\partial \rho} + s \nabla \frac{(\rho \mathcal U (\rho, s))}{\partial s} + \mathbf B \times \nabla \times \mathbf B - \nabla \cdot \left((\mu+\mu_a(\mathbf x)) \nabla \mathbf u \right) = 0 \,, \\[4mm] &\partial_t s + \nabla \cdot ( s \mathbf u ) = \frac{1}{T}\left((\mu+\mu_a(\mathbf x)) |\nabla \mathbf u|^2 + (\eta + \eta_a(\mathbf x)) |\nabla \times \mathbf B|^2\right) \,, \\[4mm] &\partial_t \mathbf B + \nabla \times ( \mathbf B \times \mathbf u ) + \nabla \times (\eta + \eta_a(\mathbf x)) \nabla \times \mathbf B = 0 \,,\end{split}\]

where the internal energy per unit mass is \(\mathcal U(\rho) = \rho^{\gamma-1} \exp(s / \rho)\), and \(\mu_a(\mathbf x)\) and \(\eta_a(\mathbf x)\) are artificial viscosity and resistivity coefficients.

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Propagators(with_viscosity: bool = True, with_resistivity: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscoResistiveMHD_with_p(with_viscosity: bool = True, with_resistivity: bool = True)[source]#

Bases: StruphyModel

Full (non-linear) visco-resistive MHD equations, with the pressure variable discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,.\]

\[\begin{split}&\partial_t \rho + \nabla \cdot ( \rho \mathbf u ) = 0 \,, \\[4mm] &\partial_t (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u) + \frac{1}{\gamma -1} \nabla p + \mathbf B \times \nabla \times \mathbf B - \nabla \cdot \left((\mu+\mu_a(\mathbf x)) \nabla \mathbf u \right) = 0 \,, \\[4mm] &\partial_t p + u \cdot \nabla p + \gamma p \nabla \cdot u = \frac{1}{(\gamma -1)}\left((\mu+\mu_a(\mathbf x)) |\nabla \mathbf u|^2 + (\eta + \eta_a(\mathbf x)) |\nabla \times \mathbf B|^2\right) \,, \\[4mm] &\partial_t \mathbf B + \nabla \times ( \mathbf B \times \mathbf u ) + \nabla \times (\eta + \eta_a(\mathbf x)) \nabla \times \mathbf B = 0 \,,\end{split}\]

and \(\mu_a(\mathbf x)\) and \(\eta_a(\mathbf x)\) are artificial viscosity and resistivity coefficients.

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Diagnostics[source]#: Bases: DiagnosticSpecies

class Propagators(with_viscosity: bool = True, with_resistivity: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscoResistiveMHD_with_q(with_viscosity: bool = True, with_resistivity: bool = True)[source]#

Bases: StruphyModel

Full (non-linear) visco-resistive MHD equations, with the q variable (square root of the pressure) discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,.\]

\[\begin{split}&\partial_t \rho + \nabla \cdot ( \rho \mathbf u ) = 0 \,, \\[4mm] &\partial_t (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u) + \frac{2q}{\gamma -1} \nabla q + \mathbf B \times \nabla \times \mathbf B - \nabla \cdot \left((\mu+\mu_a(\mathbf x)) \nabla \mathbf u \right) = 0 \,, \\[4mm] &\partial_t q + \cdot(\nabla q \mathbf u) + (\gamma/2 -1) q \nabla \cdot u = \frac{2 q}{(\gamma -1)}\left((\mu+\mu_a(\mathbf x)) |\nabla \mathbf u|^2 + (\eta + \eta_a(\mathbf x)) |\nabla \times \mathbf B|^2\right) \,, \\[4mm] &\partial_t \mathbf B + \nabla \times ( \mathbf B \times \mathbf u ) + \nabla \times (\eta + \eta_a(\mathbf x)) \nabla \times \mathbf B = 0 \,,\end{split}\]

and \(\mu_a(\mathbf x)\) and \(\eta_a(\mathbf x)\) are artificial viscosity and resistivity coefficients.

Propagators (called in sequence):

VariationalDensityEvolve

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class MHD[source]#: Bases: FluidSpecies

class Diagnostics[source]#: Bases: DiagnosticSpecies

class Propagators(with_viscosity: bool = True, with_resistivity: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscousEulerSPH(with_B0: bool = True)[source]#

Bases: StruphyModel

Euler equations discretized with smoothed particle hydrodynamics (SPH).

\[\hat u = \hat v_\textnormal{th} \,.\]

\[\begin{split}\begin{align} \partial_t \rho + \nabla \cdot (\rho \mathbf u) &= 0\,, \\[2mm] \rho(\partial_t \mathbf u + \mathbf u \cdot \nabla \mathbf u) &= - \nabla \left(\rho^2 \frac{\partial \mathcal U(\rho, S)}{\partial \rho} \right)\,, \\[2mm] \partial_t S + \mathbf u \cdot \nabla S &= 0\,, \end{align}\end{split}\]

where \(S\) denotes the entropy per unit mass. The internal energy per unit mass can be defined in two ways:

\[ \begin{align}\begin{aligned}\mathrm{"isothermal:"}\qquad &\mathcal U(\rho, S) = \kappa(S) \log \rho\,.\\\mathrm{"polytropic:"}\qquad &\mathcal U(\rho, S) = \kappa(S) \frac{\rho^{\gamma - 1}}{\gamma - 1}\,.\end{aligned}\end{align} \]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EulerFluid[source]#: Bases: ParticleSpecies

class Propagators(with_B0: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.ViscousFluid(with_viscosity: bool = True)[source]#

Bases: StruphyModel

Full (non-linear) viscous Navier-Stokes equations discretized with a variational method.

\[\hat u = \hat v_\textnormal{A}\,, \qquad \hat{\mathcal U} = \frac{\hat{\mathbf B}^2}{\hat \rho \mu_0 (\gamma-1)} \,,\qquad \hat s = \hat \rho\ \textrm{ln}\left(\frac{\hat{\mathbf B}^2}{\mu_0 (\gamma -1) \hat{\rho}}\right) \,.\]

\[\begin{split}&\partial_t \rho + \nabla \cdot ( \rho \mathbf u ) = 0 \,, \\[4mm] &\partial_t (\rho \mathbf u) + \nabla \cdot (\rho \mathbf u \otimes \mathbf u) + \rho \nabla \frac{(\rho \mathcal U (\rho, s))}{\partial \rho} + s \nabla \frac{(\rho \mathcal U (\rho, s))}{\partial s} - \nabla \cdot \left((\mu +\mu_a(\mathbf x)) \nabla \mathbf u\right) = 0 \,, \\[4mm] &\partial_t s + \nabla \cdot ( s \mathbf u ) = \frac{1}{T}\left((\mu+\mu_a(\mathbf x)) |\nabla \mathbf u|^2 \right) \,,\end{split}\]

where the internal energy per unit mass is \(\mathcal U(\rho) = \rho^{\gamma-1} \exp(s / \rho)\). and \(\mu_a(\mathbf x)\) is an artificial viscosity coefficient.

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class Fluid[source]#: Bases: FluidSpecies

class Propagators(with_viscosity: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.Vlasov[source]#

Bases: StruphyModel

Vlasov equation in static background magnetic field.

\[\hat v = \hat \Omega_\textnormal{c} \hat x\,.\]

\[\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla f + \left(\mathbf{v}\times\mathbf{B}_0 \right) \cdot \frac{\partial f}{\partial \mathbf{v}} = 0\,.\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class KineticIons[source]#: Bases: ParticleSpecies

class Propagators[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

class struphy.models.VlasovAmpereOneSpecies(with_B0: bool = True)[source]#

Bases: StruphyModel

Vlasov-Ampère equations for one species.

\[\begin{align} \hat v = c \,, \qquad \hat E = \hat B \hat v\,,\qquad \hat \phi = \hat E \hat x \,. \end{align}\]

\[\begin{split}&\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,, \\[2mm] -&\frac{\partial \mathbf{E}}{\partial t} = \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} f \, \text{d}^3 \mathbf{v}\,,\end{split}\]

with the normalization parameter

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,,\qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \qquad \hat\Omega_\textnormal{p} = \sqrt{\frac{\hat n (Ze)^2}{\epsilon_0 (A m_\textnormal{H})}} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

where \(Z=-1\) and \(A=1/1836\) for electrons. At initial time the weak Poisson equation is solved once to weakly satisfy Gauss’ law,

\[\begin{split}\begin{align} \int_\Omega \nabla \psi^\top \cdot \nabla \phi \,\textrm d \mathbf x &= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi\, (f - f_0) \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,, \\[2mm] \mathbf{E}(t=0) &= -\nabla \phi(t=0)\,. \end{align}\end{split}\]

Moreover, it is assumed that

\[\nabla \times \mathbf B_0 = \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v}\,,\]

where \(\mathbf B_0\) is the static equilibirum magnetic field.

Notes

The Control variate method for Ampère’s law is optional; in case it is enabled via the parameter file, the following system is solved:

Find \((\mathbf E, f) \in H(\textnormal{curl}) \times C^\infty\) such that

\[\begin{split}\begin{align} -\int_\Omega \mathbf F\, \cdot \, &\frac{\partial \mathbf{E}}{\partial t}\,\textrm d \mathbf x = \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \mathbf F \cdot \mathbf{v} (f - f_0) \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \mathbf F \in H(\textnormal{curl}) \,, \\[2mm] &\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon} \left( \mathbf{E} + \mathbf{v} \times \mathbf{B}_0 \right) \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,. \end{align}\end{split}\]

Propagators (called in sequence):

classmethod model_type() → Literal['Toy', 'Kinetic', 'Fluid', 'Hybrid'][source]#: Model type (Fluid, Kinetic, Hybrid, or Toy)

class EMFields[source]#: Bases: FieldSpecies

class KineticIons[source]#: Bases: ParticleSpecies

class Propagators(with_B0: bool = True)[source]#: Bases: object

allocate_helpers()[source]#: Allocate helper arrays that are needed during simulation.

generate_default_parameter_file(path=None, prompt=True)[source]#

Generate a parameter file with default options for each species, and save it to the current input path.

The default name is params_<model_name>.yml.

Parameters:

path (str) – Alternative path to getcwd()/params_MODEL.py.
prompt (bool) – Whether to prompt for overwriting the specified .yml file.

Returns:

params_path – The path of the parameter file.

Return type:

str

class struphy.models.VlasovMaxwellOneSpecies[source]#

Bases: StruphyModel

Vlasov-Maxwell equations for one species.

\[\begin{align} \hat v = c \,, \qquad \hat E = \hat B \hat v\,,\qquad \hat \phi = \hat E \hat x \,. \end{align}\]

\[\begin{split}&\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon} \left( \mathbf{E} + \mathbf{v} \times \left( \mathbf{B} + \mathbf{B}_0 \right) \right) \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,, \\[2mm] -&\frac{\partial \mathbf{E}}{\partial t} + \nabla \times \mathbf B = \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} f \, \text{d}^3 \mathbf{v}\,, \\[2mm] &\frac{\partial \mathbf{B}}{\partial t} + \nabla \times \mathbf{E} = 0 \,,\end{split}\]

with the normalization parameters

\[\alpha = \frac{\hat \Omega_\textnormal{p}}{\hat \Omega_\textnormal{c}}\,,\qquad \varepsilon = \frac{1}{\hat \Omega_\textnormal{c} \hat t} \,,\qquad \textnormal{with} \qquad \hat\Omega_\textnormal{p} = \sqrt{\frac{\hat n (Ze)^2}{\epsilon_0 (A m_\textnormal{H})}} \,,\qquad \hat \Omega_{\textnormal{c}} = \frac{(Ze) \hat B}{(A m_\textnormal{H})}\,,\]

where \(Z=-1\) and \(A=1/1836\) for electrons. At initial time the weak Poisson equation is solved once to weakly satisfy Gauss’ law,

\[\begin{split}\begin{align} \int_\Omega \nabla \psi^\top \cdot \nabla \phi \,\textrm d \mathbf x &= \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \psi\, (f - f_0) \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \psi \in H^1\,, \\[2mm] \mathbf{E}(t=0) &= -\nabla \phi(t=0)\,. \end{align}\end{split}\]

Moreover, it is assumed that

\[\nabla \times \mathbf B_0 = \frac{\alpha^2}{\varepsilon} \int_{\mathbb{R}^3} \mathbf{v} f_0 \, \text{d}^3 \mathbf{v}\,,\]

where \(\mathbf B_0\) is the static equilibirum magnetic field.

Notes

The Control variate method for Ampère’s law is optional; in case it is enabled via the parameter file, the following system is solved:

Find \((\mathbf E, \tilde{\mathbf B}, f) \in H(\textnormal{curl}) \times H(\textnormal{div}) \times C^\infty\) such that

\[\begin{split}\begin{align} -\int_\Omega \mathbf F\, \cdot \, &\frac{\partial \mathbf{E}}{\partial t}\,\textrm d \mathbf x + \int_\Omega \nabla \times \mathbf{F} \cdot \tilde{\mathbf B}\,\textrm d \mathbf x = \frac{\alpha^2}{\varepsilon} \int_\Omega \int_{\mathbb{R}^3} \mathbf F \cdot \mathbf{v} (f - f_0) \, \text{d}^3 \mathbf{v}\,\textrm d \mathbf x \qquad \forall \ \mathbf F \in H(\textnormal{curl}) \,, \\[2mm] &\frac{\partial \tilde{\mathbf B}}{\partial t} + \nabla \times \mathbf{E} = 0 \,, \\[2mm] &\frac{\partial f}{\partial t} + \mathbf{v} \cdot \, \nabla f + \frac{1}{\varepsilon}\Big[ \mathbf{E} + \mathbf{v} \times (\mathbf{B}_0 + \tilde{\mathbf B}) \Big] \cdot \frac{\partial f}{\partial \mathbf{v}} = 0 \,, \end{align}\end{split}\]

where \(\tilde{\mathbf B} = \mathbf B - \mathbf B_0\) denotes the magnetic perturbation.

Propagators (called in sequence):