sapphirepp::VFP Namespace Reference

Namespace for the Vlasov-Fokker-Planck module. More...

Classes
class	InitialValueFunction
	Initial condition. More...
class	BoundaryValueFunction
	Inflow boundary condition. More...
class	ScatteringFrequency
	Scattering frequency. More...
class	Source
	Source term. More...
class	MagneticField
	Magnetic field. More...
class	BackgroundVelocityField
	Background velocity field. More...
class	DebugInputFunctions
	This class collects the user-defined input functions to output them as debug information. More...
class	NumericalFlux
	Class to compute the numerical fluxes in the VFP equation. More...
class	ParticleVelocity
	A function to compute the particle velocity at a point in reduced phase space \( (\mathbf{x}, p) \). More...
class	ParticleGamma
	Function to compute the particle gamma factor \( \gamma \) at a point in reduced phase space \( (\mathbf{x}, p) \). More...
class	PDESystem
	Calculate the matrices of the PDE system. More...
class	ProbeLocation
	PostProcessor unit to probe location in reduced phase space and reconstruct the phase space distribution. More...
struct	ReferenceValues
	Reference values for the VFP equation. More...
class	VFPParameters
	Parameters class for the VFP module. More...
class	VFPSolver
	This class solves the Vlasov-Fokker-Planck equation. More...

Enumerations
enum class	VFPFlags : std::uint16_t {   none = 0 , time_evolution = 1 << 0 , spatial_advection = 1 << 1 , collision = 1 << 2 ,   rotation = 1 << 3 , time_independent_fields = 1 << 4 , momentum = 1 << 5 , linear_p = 1 << 6 ,   source = 1 << 7 , time_independent_source = 1 << 8 , scaled_distribution_function = 1 << 9 , local_lax_friedrichs_flux = 1 << 10 ,   radiation_reaction = 1 << 11 }
	Flags to activate the different terms of the VFP equation. More...
enum class	BoundaryConditions : std::uint8_t {   continuous , zero_inflow , reflective , inflow ,   periodic }
	Boundary conditions for the VFP equation. More...
enum class	TimeSteppingMethod : std::uint8_t {   crank_nicolson , forward_euler , backward_euler , erk4 ,   lserk4 }
	Time stepping methods for the VFP equation. More...
enum class	GridType : std::uint8_t { hypercube , shock , file }
	Grid generation types for the VFP equation. More...

Functions
template<typename StreamType>
StreamType &	operator<< (StreamType &os, const ReferenceValues &reference_values)
	Output reference values to a stream.
constexpr VFPFlags	operator| (VFPFlags f1, VFPFlags f2)
	Implement the bitwise OR operator for the VFPFlags.
constexpr VFPFlags	operator& (VFPFlags f1, VFPFlags f2)
	Implement the bitwise AND operator for the VFPFlags.
template<typename StreamType>
StreamType &	operator<< (StreamType &os, VFPFlags f)
	Print the VFPFlags to a stream.

Variables
constexpr unsigned int	dimension = 2
constexpr VFPFlags	vfp_flags

Detailed Description

Namespace for the Vlasov-Fokker-Planck module.

Enumeration Type Documentation

BoundaryConditions

enum class sapphirepp::VFP::BoundaryConditions : std::uint8_t

strong

Boundary conditions for the VFP equation.

Enumerator
continuous	Continuous. The distribution function at the boundary is continuously extrapolated from the interior.
zero_inflow	Zero inflow. Only outflow, no inflow at the boundary.
reflective	Reflective. The particles velocity is reflected at the boundary.
inflow	Inflow. Inflow plus outflow. Inflow at specified boundary is determined by a boundary value function.
periodic	Periodic boundary conditions. Has to be set on both sides of the boundary.

GridType

enum class sapphirepp::VFP::GridType : std::uint8_t

strong

Grid generation types for the VFP equation.

Enumerator
hypercube	Create a hypercube grid.
shock	Create a grid for a shock problem in x-direction.
file	Use a grid that is read from a file.

TimeSteppingMethod

enum class sapphirepp::VFP::TimeSteppingMethod : std::uint8_t

strong

Time stepping methods for the VFP equation.

Enumerator
crank_nicolson	Implicit Crank-Nicolson method.
forward_euler	Explicit Euler method.
backward_euler	Implicit Euler method.
erk4	Fourth order Runge-Kutta method.
lserk4	Low storage fourth order Runge-Kutta method.

VFPFlags

enum class sapphirepp::VFP::VFPFlags : std::uint16_t

strong

Flags to activate the different terms of the VFP equation.

We split the VFP equation into different terms,

\begin{align} \frac{\partial f}{\partial t} & \quad & \text{(time-evolution term)} \\ & + (\mathbf{u} + \mathbf{v}) \cdot \nabla_{x} f & \text{(spatial advection term)} \\ & - \gamma m \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} \cdot \nabla_{p}f - \mathbf{p} \cdot\nabla_{x} \mathbf{u}\cdot \nabla_{p} f & \text{(momentum term)} \\ & + q \mathbf{v} \cdot \left( \mathbf{B} \times \nabla_{p} f \right) & \text{(rotation term)} \\ & + \nabla_{p} \cdot \left( \sigma \Big( \mathbf{B}\,(\mathbf{B}\cdot\boldsymbol{\beta}) + \gamma^{2}\,\boldsymbol{\beta}\, \big((\boldsymbol{\beta}\cdot\mathbf{B})^{2} - B^{2}\big) \Big)\, f \right) & \text{(radiation reaction term)} \\ = & \frac{\nu}{2} \Delta_{\theta, \varphi} f & \text{(collision term)} \\ & + S \,. & \text{(source term)} \\ \end{align}

These terms can be individually activated or deactivated.

Enumerator
time_evolution	Activate the time-evolution term \( \frac{\partial f}{\partial t} \)
spatial_advection	Activate the spatial advection term \( (\mathbf{u} + \mathbf{v}) \cdot \nabla_x f \)
collision	Activate the collision term \( \frac{\nu}{2} \Delta_{\theta, \varphi} f \)
rotation	Activate the rotation term, i.e. the magnetic field \( q \mathbf{v} \cdot \left( \mathbf{B} \times \nabla_{p} f \right) \)
time_independent_fields	If the fields \( \mathbf{u} \) and \(\mathbf{B} \) are time independent, this flag should be used. This significantly improves performance, as it requires the dg_matrix to be assembled only once. By default the fields are assumed to be time dependent.
momentum	Activate the momentum term \( \left( \gamma m \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} + \mathbf{p} \cdot\nabla_{x} \mathbf{u} \right) \cdot \nabla_{p} f \)
linear_p	Use a linear momentum variable, \( p \). By default a logarithmic momentum variable, \( \ln(p) \), is used.
source	Activate the source term \( S(\mathbf{x}, \mathbf{p}, t) \)
time_independent_source	If the source \( S \) is time independent, this flag should be used. This significantly improves performance, as it requires the system_rhs to be assembled only once. By default the source is assumed to be time dependent.
scaled_distribution_function	Use a scaled in distribution function, \( g = p^s f \). The exponent is fixed to \( s = 3 \), i.e. \( g = p^3 f \).
local_lax_friedrichs_flux	Use the local Lax-Friedrichs flux instead of the default Upwind flux. The Upwind flux is much more memory heavy but is more robust. The local Lax-Friedrichs flux is less robust but requires less memory.
radiation_reaction	Activate the radiation reaction term \( + \nabla_{p} \cdot \left( \sigma \Big( \mathbf{B}\,(\mathbf{B}\cdot\boldsymbol{\beta}) + \gamma^{2}\,\boldsymbol{\beta} \left((\boldsymbol{\beta}\cdot\mathbf B)^{2} - B^{2}\right) \Big)\, f \right) \)

Function Documentation

operator&()

VFPFlags sapphirepp::VFP::operator& ( VFPFlags f1, VFPFlags f2 )

constexpr

Implement the bitwise AND operator for the VFPFlags.

Parameters

f1	First VFPFlags
f2	Second VFPFlags

Returns

constexpr VFPFlags f1 & f2

operator<<() [1/2]

template<typename StreamType>

StreamType & sapphirepp::VFP::operator<<	(	StreamType &	os,
		const ReferenceValues &	reference_values )

Output reference values to a stream.

Template Parameters

StreamType

Type of the stream

Parameters

os	Output stream
reference_values	Reference values

Returns

StreamType& os

operator<<() [2/2]

template<typename StreamType>

StreamType & sapphirepp::VFP::operator<< ( StreamType & os, VFPFlags f )

inline

Print the VFPFlags to a stream.

Template Parameters

StreamType

Type of the output stream

Parameters

os	Output stream
f	VFPFlags

Returns

StreamType& os

operator|()

VFPFlags sapphirepp::VFP::operator| ( VFPFlags f1, VFPFlags f2 )

constexpr

Implement the bitwise OR operator for the VFPFlags.

Parameters

f1	First VFPFlags
f2	Second VFPFlags

Returns

constexpr VFPFlags f1 | f2

Variable Documentation

dimension

unsigned int sapphirepp::VFP::dimension = 2

constexpr

[Dimension] Specify reduced phase space dimension \( (\mathbf{x}, p) \)

vfp_flags

VFPFlags sapphirepp::VFP::vfp_flags

constexpr

Initial value:

= VFPFlags::time_evolution |          
  VFPFlags::spatial_advection |       
  VFPFlags::momentum |                
  VFPFlags::collision |               
  VFPFlags::source |                  
  VFPFlags::time_independent_fields | 
  VFPFlags::time_independent_source

[Dimension] [VFP Flags] Specify which terms of the VFP equation should be active