BorisStepper

Electromagnetic particle stepper method implementing the Boris Algorithm.

Verification of the Boris stepper implementation is detailed here.

In magnetized (or electromagnetic) PIC simulations, the de facto standard particle stepping algorithm is commonly known as the Boris algorithm Boris and others (1970), Birdsall and Langdon (1991), and Qin et al. (2013). This algorithm is similar to a Leap Frog method and has second order accuracy in time when solving the equations of motion for a charged particle, given by

$(1)var element = document.getElementById("moose-equation-82fd77f1-de1d-4dbc-b516-368fa3f7509d");katex.render(" \\frac{\\partial \\vec{r}}{\\partial t} = \\vec{v}", element, {displayMode:true,throwOnError:false});$

and

$(2)var element = document.getElementById("moose-equation-3c46885f-6691-4e3d-b5d2-6eacc2301ebc");katex.render(" \\frac{\\partial \\vec{v}}{\\partial t} = \\frac{q}{m} \\left[ \\vec{E} + \\vec{v} \\times \\vec{B} \\right]", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-c26c2a91-a964-410b-89f6-fc916edb97cf");katex.render("q", element, {displayMode:false,throwOnError:false});$ is the particle's charge, $var element = document.getElementById("moose-equation-9217e043-506b-493a-a4e4-6ca3a7744e20");katex.render("m", element, {displayMode:false,throwOnError:false});$ is the particle's mass, and $var element = document.getElementById("moose-equation-1d3b1bb3-c9c1-4411-bf46-8c12856f29a6");katex.render("\\vec{E}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-824e0250-1f7d-4590-9bbd-401b7f322af7");katex.render("\\vec{B}", element, {displayMode:false,throwOnError:false});$ are the electric and magnetic fields that the particle is subject to, respectively.

In the Boris algorithm, \cref{eq:pos,eq:vel} are discretized with a central difference scheme and the acceleration due to the electric field and magnetic field are separated. First, half of the impulse due to the electric field is applied to the particle, as

$(3)var element = document.getElementById("moose-equation-d90f424c-4895-4f86-8ca1-e9d524f4df86");katex.render(" \\vec{v}^{\\,-} = \\vec{v}_{n} + \\frac{q}{m} \\vec{E}_n \\frac{\\Delta t}{2}", element, {displayMode:true,throwOnError:false});$

where $var element = document.getElementById("moose-equation-b603cde0-deb9-4ab1-9aca-73a1c3949fb7");katex.render("\\vec{v}^{\\,-}", element, {displayMode:false,throwOnError:false});$ is an intermediate particle velocity, $var element = document.getElementById("moose-equation-2e3a4d22-19ca-48bb-ba29-b06aafda5739");katex.render("\\vec{v}_{n}", element, {displayMode:false,throwOnError:false});$ is the particle velocity at step $var element = document.getElementById("moose-equation-ae2e623e-86eb-4a3f-948c-0a67bf494cfa");katex.render("n", element, {displayMode:false,throwOnError:false});$ , and $var element = document.getElementById("moose-equation-dd3c7e5f-ed93-4183-be39-bf9703c9c456");katex.render("\\vec{E}_{n}", element, {displayMode:false,throwOnError:false});$ is the electric field at step $var element = document.getElementById("moose-equation-f1002a55-6654-4b57-bec3-f6dcc4b39b18");katex.render("n", element, {displayMode:false,throwOnError:false});$ . The velocity of the particle after rotation due to the magnetic field is derived as

$(4)var element = document.getElementById("moose-equation-5f456fc2-c7e7-4e21-be6b-1af50f54a45f");katex.render(" \\vec{v}^{\\,+} = \\vec{v}^{\\,-} + \\vec{v}^{\\,'} \\times \\vec{s},", element, {displayMode:true,throwOnError:false});$

with

$(5)var element = document.getElementById("moose-equation-cec0576c-418d-4c47-8d1e-edf5d88ec517");katex.render(" \\vec{v}^{\\,'} = \\vec{v}^{\\,-} + \\vec{v}^{\\,-} \\times \\vec{l}", element, {displayMode:true,throwOnError:false});$

where

$(6)var element = document.getElementById("moose-equation-b279dac0-662f-4712-836a-f260c4f94b3c");katex.render(" \\vec{l} = \\frac{q}{m} \\vec{B}_n \\Delta t,", element, {displayMode:true,throwOnError:false});$

which accounts for the effect of $var element = document.getElementById("moose-equation-94aa7f7c-41ed-4b21-b6fe-5b62fc7a60a2");katex.render("\\vec{B}_n", element, {displayMode:false,throwOnError:false});$ , the magnetic field at step $var element = document.getElementById("moose-equation-289c8da2-8124-4c15-a473-a36888ded598");katex.render("n", element, {displayMode:false,throwOnError:false});$ . $var element = document.getElementById("moose-equation-0303a3f2-f28f-4604-a2cf-22b14e6339cb");katex.render("\\vec{s}", element, {displayMode:false,throwOnError:false});$ is defined as

$(7)var element = document.getElementById("moose-equation-47e2dcd1-6e91-4d2a-8344-886c1c9f01a2");katex.render(" \\vec{s} = \\frac{2 \\vec{l}}{ 1 + \\vec{l} \\cdot \\vec{l} }.", element, {displayMode:true,throwOnError:false});$

Finally, the rotation due to the presence of the magnetic field is then applied with

$(8)var element = document.getElementById("moose-equation-54d1774f-37dd-4848-8b0e-a8dc2af78798");katex.render(" \\frac{ \\vec{v}^{\\,+} - \\vec{v}^{\\,-} }{ \\Delta t} = \\frac{q}{m} \\left( \\vec{v}^{\\,+} + \\vec{v}^{\\,-} \\right) \\times \\vec{B}_n,", element, {displayMode:true,throwOnError:false});$

and the final impulse due to the electric field is then applied to the particle using

$(9)var element = document.getElementById("moose-equation-7e3771c3-c358-42a9-b889-e6734aacd811");katex.render(" \\vec{v}_{n+1} = \\vec{v}^{\\,+} + \\frac{q}{m} \\vec{E}_n \\frac{\\Delta t}{2}.", element, {displayMode:true,throwOnError:false});$

The implementation of the Boris algorithm was verified using several single particle motion tests: constant electric field ((test/tests/userobjects/particle_stepper/boris_stepper/parallel_acceleration.i), (test/tests/userobjects/particle_stepper/boris_stepper/projectile_motion.i)), cyclotron motion ((test/tests/userobjects/particle_stepper/boris_stepper/cyclotron_motion.i)), and $var element = document.getElementById("moose-equation-2fc907ae-eb45-4a1a-a6a8-349b74020624");katex.render("\\vec{E} \\times \\vec{B}", element, {displayMode:false,throwOnError:false});$ drift motion ((test/tests/userobjects/particle_stepper/boris_stepper/e_cross_b.i)).

Example Input Syntax

[UserObjects<<<{"href": "../../syntax/UserObjects/index.html"}>>>]
  [velocity_initializer]
    type = ConstantVelocityInitializer<<<{"description": "Provides particles with velocities sampled from a user provided list of velocities.", "href": "ConstantVelocityInitializer.html"}>>>

    velocities<<<{"description": "The velocites which will be cycled through when initializing particles."}>>> = '1 0 0'
  []

  [stepper]

    type = BorisStepper<<<{"description": "Electromagnetic particle stepper method implementing the Boris Algorithm.", "href": "BorisStepper.html"}>>>
    efield_components<<<{"description": "A list of 3 variables which represent the 3 components of the electric field"}>>> = 'Ex Ey Ez'
    bfield_components<<<{"description": "A list of 3 variables which represent the 3 components of the electric field"}>>> = 'Bx By Bz'
  []

  [particle_initializer]
    type = TestPlacedParticleInitializer
    velocity_initializer = 'velocity_initializer'

    start_points = '0 1 0'
    mass = 1
    weight = 1
    charge = 1
  []

  [study]
    type = TestInitializedPICStudy
    stepper = stepper
    particle_initializer = particle_initializer
    use_custom_rayids = false
    always_cache_traces = true
    data_on_cache_traces = true
    execute_on = 'TIMESTEP_BEGIN'
    ray_kernel_coverage_check = false
  []
[]

Input Parameters

bfield_componentsA list of 3 variables which represent the 3 components of the electric field
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:A list of 3 variables which represent the 3 components of the electric field
efield_componentsA list of 3 variables which represent the 3 components of the electric field
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:A list of 3 variables which represent the 3 components of the electric field

Required Parameters

allow_duplicate_execution_on_initialFalseIn the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
Default:False
C++ Type:bool
Controllable:No
Description:In the case where this UserObject is depended upon by an initial condition, allow it to be executed twice during the initial setup (once before the IC and again after mesh adaptivity (if applicable).
execute_onTIMESTEP_ENDThe list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
Default:TIMESTEP_END
C++ Type:ExecFlagEnum
Options:XFEM_MARK, NONE, INITIAL, LINEAR, LINEAR_CONVERGENCE, NONLINEAR, NONLINEAR_CONVERGENCE, POSTCHECK, TIMESTEP_END, TIMESTEP_BEGIN, MULTIAPP_FIXED_POINT_END, MULTIAPP_FIXED_POINT_BEGIN, MULTIAPP_FIXED_POINT_CONVERGENCE, FINAL, CUSTOM, TRANSFER
Controllable:No
Description:The list of flag(s) indicating when this object should be executed. For a description of each flag, see https://mooseframework.inl.gov/source/interfaces/SetupInterface.html.
execution_order_group0Execution order groups are executed in increasing order (e.g., the lowest number is executed first). Note that negative group numbers may be used to execute groups before the default (0) group. Please refer to the user object documentation for ordering of user object execution within a group.
Default:0
C++ Type:int
Controllable:No
Description:Execution order groups are executed in increasing order (e.g., the lowest number is executed first). Note that negative group numbers may be used to execute groups before the default (0) group. Please refer to the user object documentation for ordering of user object execution within a group.
force_postauxFalseForces the UserObject to be executed in POSTAUX
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in POSTAUX
force_preauxFalseForces the UserObject to be executed in PREAUX
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in PREAUX
force_preicFalseForces the UserObject to be executed in PREIC during initial setup
Default:False
C++ Type:bool
Controllable:No
Description:Forces the UserObject to be executed in PREIC during initial setup

Execution Scheduling Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:Yes
Description:Set the enabled status of the MooseObject.
use_displaced_meshFalseWhether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.
Default:False
C++ Type:bool
Controllable:No
Description:Whether or not this object should use the displaced mesh for computation. Note that in the case this is true but no displacements are provided in the Mesh block the undisplaced mesh will still be used.

Advanced Parameters

prop_getter_suffixAn optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
C++ Type:MaterialPropertyName
Unit:(no unit assumed)
Controllable:No
Description:An optional suffix parameter that can be appended to any attempt to retrieve/get material properties. The suffix will be prepended with a '_' character.
use_interpolated_stateFalseFor the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.
Default:False
C++ Type:bool
Controllable:No
Description:For the old and older state use projected material properties interpolated at the quadrature points. To set up projection use the ProjectedStatefulMaterialStorageAction.

Material Property Retrieval Parameters

Input Files

(test/tests/userobjects/particle_stepper/boris_stepper/boris_base.i)

References

Charles K. Birdsall and A. Bruce Langdon. Plasma physics via computer simulation. Adam Hilger, 1991.

@book{Birdsall_Langdon_1991,
    author = "Birdsall, Charles K. and Langdon, A. Bruce",
    place = "Bristol, NY",
    title = "Plasma physics via computer simulation",
    publisher = "Adam Hilger",
    year = "1991"
}

Jay P Boris and others. Relativistic plasma simulation-optimization of a hybrid code. In Proc. Fourth Conf. Num. Sim. Plasmas, 3–67. 1970.

@inproceedings{boris1970relativistic,
    author = "Boris, Jay P and others",
    title = "Relativistic plasma simulation-optimization of a hybrid code",
    booktitle = "Proc. Fourth Conf. Num. Sim. Plasmas",
    pages = "3--67",
    year = "1970"
}

Hong Qin, Shuangxi Zhang, Jianyuan Xiao, Jian Liu, Yajuan Sun, and William M Tang. Why is boris algorithm so good? Physics of Plasmas, 2013.

@article{qin2013boris,
    author = "Qin, Hong and Zhang, Shuangxi and Xiao, Jianyuan and Liu, Jian and Sun, Yajuan and Tang, William M",
    title = "Why is Boris algorithm so good?",
    journal = "Physics of Plasmas",
    volume = "20",
    number = "8",
    year = "2013",
    publisher = "AIP Publishing"
}

(test/tests/userobjects/particle_stepper/boris_stepper/parallel_acceleration.i)

# The boris_base.i file sets up everything that the simulation needs to utilize
# the BorisStepper and sample the proper field variables.
!include boris_base.i
# This file gives the field variables the proper state to demonstrate a simple case of
# an impulse parallel to the particle velocity.

[Mesh/gmg]
  nx = 120
  ny = 20
  xmax = 120
  ymax = 20
[]

[Functions]
  [E_x_ic]
    expression = '5e-7'
  []
  [E_y_ic]
    expression = '0'
  []
  [E_z_ic]
    expression = '0'
  []
  [B_x_ic]
    expression = '0'
  []
  [B_y_ic]
    expression = '0'
  []
  [B_z_ic]
    expression = '0'
  []
[]


[UserObjects]
  [velocity_initializer]
    velocities = '0 0 0'
  []

  [particle_initializer]
    start_points = '0 10.5 0'
    mass = 9.1093837015e-31
    charge = 1.602176634e-19
  []
[]

[Executioner]
  dt = 1e-4
[]

(test/tests/userobjects/particle_stepper/boris_stepper/projectile_motion.i)

# The boris_base.i file sets up everything that the simulation needs to utilize
# the BorisStepper and sample the proper field variables.
!include boris_base.i
# This file gives the field variables the proper state to demonstrate a simple case of
# projectile motion.

[Mesh/gmg]
  nx = 25
  ny = 7
  xmax = 22
  ymax = 7
[]

[Functions]
  [E_x_ic]
    expression = '0'
  []
  [E_y_ic]
    expression = '-9.81'
  []
  [E_z_ic]
    expression = '0'
  []
  [B_x_ic]
    expression = '0'
  []
  [B_y_ic]
    expression = '0'
  []
  [B_z_ic]
    expression = '0'
  []
[]

[UserObjects]
  [velocity_initializer]
    velocities = '10 10 0'
  []

  [particle_initializer]
    start_points = '0 0 0'
    mass = 1
    charge = 1
  []
[]


[Executioner]
  dt = 1e-2
[]

(test/tests/userobjects/particle_stepper/boris_stepper/cyclotron_motion.i)

# The boris_base.i file sets up everything that the simulation needs to utilize
# the BorisStepper and sample the proper field variables.
!include boris_base.i
# This file gives the field variables the proper state for the cyclotron motion case,
# and creates a particle with the proper initial conditions.

[Mesh/gmg]
  nx = 5
  ny = 5
  xmin = -2
  xmax = 2
  ymin = -2
  ymax = 2
[]

[Functions]
  [E_x_ic]
    expression = '0'
  []
  [E_y_ic]
    expression = '0'
  []
  [E_z_ic]
    expression = '0'
  []
  [B_x_ic]
    expression = '0'
  []
  [B_y_ic]
    expression = '0'
  []
  [B_z_ic]
    expression = '1'
  []
[]

[UserObjects]
  [velocity_initializer]
    velocities = '1 0 0'
  []

  [particle_initializer]
    start_points = '0 1 0'
    mass = 1
    weight = 1
    charge = 1
  []
[]

[Executioner]
  dt = 1e-1
[]

(test/tests/userobjects/particle_stepper/boris_stepper/e_cross_b.i)

# The boris_base.i file sets up everything that the simulation needs to utilize
# the BorisStepper and sample the proper field variables.
!include boris_base.i
# This file gives the field variables the proper state to demonstrate ExB drift.

[Mesh/gmg]
  nx = 20
  ny = 20
  xmin = -2
  xmax = 30
  ymin = -20
  ymax = 20
[]

[Functions]
  [E_x_ic]
    expression = '0.05'
  []
  [E_y_ic]
    expression = '0'
  []
  [E_z_ic]
    expression = '0'
  []
  [B_x_ic]
    expression = '0'
  []
  [B_y_ic]
    expression = '0'
  []
  [B_z_ic]
    expression = '1'
  []
[]

[UserObjects]
  [velocity_initializer]
    velocities = '1 0 0'
  []

  [particle_initializer]
    start_points = '0 1 0'
    mass = 1
    weight = 1
    charge = 1
  []
[]

[Executioner]
  dt = 1e-1
[]

(test/tests/userobjects/particle_stepper/boris_stepper/cyclotron_motion.i)

# The boris_base.i file sets up everything that the simulation needs to utilize
# the BorisStepper and sample the proper field variables.
!include boris_base.i
# This file gives the field variables the proper state for the cyclotron motion case,
# and creates a particle with the proper initial conditions.

[Mesh/gmg]
  nx = 5
  ny = 5
  xmin = -2
  xmax = 2
  ymin = -2
  ymax = 2
[]

[Functions]
  [E_x_ic]
    expression = '0'
  []
  [E_y_ic]
    expression = '0'
  []
  [E_z_ic]
    expression = '0'
  []
  [B_x_ic]
    expression = '0'
  []
  [B_y_ic]
    expression = '0'
  []
  [B_z_ic]
    expression = '1'
  []
[]

[UserObjects]
  [velocity_initializer]
    velocities = '1 0 0'
  []

  [particle_initializer]
    start_points = '0 1 0'
    mass = 1
    weight = 1
    charge = 1
  []
[]

[Executioner]
  dt = 1e-1
[]

(test/tests/userobjects/particle_stepper/boris_stepper/boris_base.i)

# The stepper base file provides the basic objects that are required for both
# the LeapFrogStepper and the BorisSteper tests
!include ../stepper_base.i
# The rest of this file adds the AuxVariables for the magnetic field, sets type of
# the particle stepper to be the BorisStepper and tells the stepper to sample the
# proper variables for the electric and magnetic fields.

[AuxVariables]
  [Bx]
  []
  [By]
  []
  [Bz]
  []
[]

[Functions]
  [B_x_ic]
    type = ParsedFunction
  []
  [B_y_ic]
    type = ParsedFunction
  []
  [B_z_ic]
    type = ParsedFunction
  []
[]

[ICs]
  [Bx_ic]
    type = FunctionIC
    variable = Bx
    function = B_x_ic
  []
  [By_ic]
    type = FunctionIC
    variable = By
    function = B_y_ic
  []
  [Bz_ic]
    type = FunctionIC
    variable = Bz
    function = B_z_ic
  []
[]

[UserObjects]
  [stepper]
    type = BorisStepper
    efield_components = 'Ex Ey Ez'
    bfield_components = 'Bx By Bz'
  []
[]

Input Parameters
Input Files
References