LevelSetAdvection

The level set equation is typically defined as below. As shown in this equation, the LevelSetAdvection kernel implements the advection portion of the equation.

$var element = document.getElementById("moose-equation-4e9af30e-4549-459f-a7ad-816d57cfc8bc");katex.render("\\frac{\\partial u}{\\partial t} + \\underbrace{\\vec{v} \\cdot \\nabla u}_{\\textrm{LevelSetAdvection}} = 0,", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ where $var element = document.getElementById("moose-equation-d22f07f1-db83-4834-ae3e-17ac7a976808");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is the level set variable, $var element = document.getElementById("moose-equation-6b0c5b66-42a6-44ec-bb55-fc874df1fa3d");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is time, and $var element = document.getElementById("moose-equation-4dd35006-0f07-4eff-b434-0985c6ea74d5");katex.render("\\vec{v}", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ is a known velocity field that advects the level set variable.

The weak form of this portion of the equation is defined as: find $var element = document.getElementById("moose-equation-862be5da-08c8-454a-8c1f-35b4d2481600");katex.render("u_h", element, {displayMode:false,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$ such that:

$var element = document.getElementById("moose-equation-6ac55b98-ca5a-45f4-8730-d404dab67ac8");katex.render("(\\Psi_i, \\vec{v}\\cdot\\nabla u_h) = 0.", element, {displayMode:true,throwOnError:false,macros:{"\\pf":"\\frac{\\partial #1}{\\partial #2}"}});$

Example Syntax

This kernel is utilized from within the Kernels block in conjunction with a time derivative kernel (TimeDerivative).

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]

  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

/opt/civet/build_0/moose/modules/level_set/test/tests/reinitialization/master.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
  nx = 8
  ny = 8
  uniform_refine = 3 #1/64
[]

[AuxVariables]
  [./vel_x]
    initial_condition = 1
  [../]
  [./vel_y]
    initial_condition = 1
  [../]
[]

[Variables]
  [./phi]
  [../]
[]

[Functions]
  [./phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.05
    center = '0.5 0.5 0'
    radius = 0.15
  [../]
[]

[BCs]
  [./Periodic]
    [./all]
      variable = phi
      auto_direction = 'x y'
    [../]
  [../]
[]

[ICs]
  [./phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]

  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

[Postprocessors]

  [./area]
    type = LevelSetVolume
    threshold = 0.5
    variable = phi
    location = outside
    execute_on = 'initial timestep_end'
  [../]

  [./cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial'
  [../]

[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  end_time = 1
  nl_rel_tol = 1e-12
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  [./TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 1
  [../]

[]

[MultiApps]
  [./reinit]
    type = LevelSetReinitializationMultiApp
    input_files = 'reinit.i'
    execute_on = 'timestep_end'
  [../]
[]

[Transfers]
  [./to_sub]
    type = MultiAppCopyTransfer
    variable = phi
    source_variable = phi
    direction = to_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  [../]

  [./to_sub_init]
    type = MultiAppCopyTransfer
    variable = phi_0
    source_variable = phi
    direction = to_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  [../]

  [./from_sub]
    type = MultiAppCopyTransfer
    variable = phi
    source_variable = phi
    direction = from_multiapp
    multi_app = reinit
    execute_on = timestep_end
  [../]
[]

[Outputs]
  exodus = true
  csv = true
[]

Input Parameters

variableThe name of the variable that this Kernel operates on
C++ Type:NonlinearVariableName
Options:
Description:The name of the variable that this Kernel operates on

Required Parameters

blockThe list of block ids (SubdomainID) that this object will be applied
C++ Type:std::vector
Options:
Description:The list of block ids (SubdomainID) that this object will be applied
displacementsThe displacements
C++ Type:std::vector
Options:
Description:The displacements
velocity_xThe variable containing the x-component of the velocity front.
C++ Type:std::vector
Options:
Description:The variable containing the x-component of the velocity front.
velocity_yThe variable containing the y-component of the velocity front.
C++ Type:std::vector
Options:
Description:The variable containing the y-component of the velocity front.
velocity_zThe variable containing the z-component of the velocity front.
C++ Type:std::vector
Options:
Description:The variable containing the z-component of the velocity front.

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector
Options:
Description:Adds user-defined labels for accessing object parameters via control logic.
diag_save_inThe name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector
Options:
Description:The name of auxiliary variables to save this Kernel's diagonal Jacobian contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Options:
Description:Set the enabled status of the MooseObject.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Options:
Description:Determines whether this object is calculated using an implicit or explicit form
save_inThe name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
C++ Type:std::vector
Options:
Description:The name of auxiliary variables to save this Kernel's residual contributions to. Everything about that variable must match everything about this variable (the type, what blocks it's on, etc.)
seed0The seed for the master random number generator
Default:0
C++ Type:unsigned int
Options:
Description:The seed for the master random number generator
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
Options:
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

extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector
Options:
Description:The extra tags for the matrices this Kernel should fill
extra_vector_tagsThe extra tags for the vectors this Kernel should fill
C++ Type:std::vector
Options:
Description:The extra tags for the vectors this Kernel should fill
matrix_tagssystemThe tag for the matrices this Kernel should fill
Default:system
C++ Type:MultiMooseEnum
Options:nontime system
Description:The tag for the matrices this Kernel should fill
vector_tagsnontimeThe tag for the vectors this Kernel should fill
Default:nontime
C++ Type:MultiMooseEnum
Options:nontime time
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

Input Files

modules/level_set/test/tests/reinitialization/master.i
modules/level_set/examples/rotating_circle/circle_rotate_master.i
modules/level_set/examples/rotating_circle/circle_rotate_supg.i
modules/level_set/examples/vortex/vortex_reinit.i
modules/level_set/examples/vortex/vortex_supg.i
modules/level_set/examples/rotating_circle/circle_rotate.i
modules/level_set/test/tests/verification/1d_level_set_supg_mms/1d_level_set_supg_mms.i
modules/level_set/test/tests/kernels/advection/advection_mms.i
modules/level_set/examples/circle/circle_16.i
modules/level_set/test/tests/verification/1d_level_set_mms/level_set_mms.i
modules/level_set/examples/vortex/vortex.i

modules/level_set/test/tests/reinitialization/master.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
  nx = 8
  ny = 8
  uniform_refine = 3 #1/64
[]

[AuxVariables]
  [./vel_x]
    initial_condition = 1
  [../]
  [./vel_y]
    initial_condition = 1
  [../]
[]

[Variables]
  [./phi]
  [../]
[]

[Functions]
  [./phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.05
    center = '0.5 0.5 0'
    radius = 0.15
  [../]
[]

[BCs]
  [./Periodic]
    [./all]
      variable = phi
      auto_direction = 'x y'
    [../]
  [../]
[]

[ICs]
  [./phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]

  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

[Postprocessors]

  [./area]
    type = LevelSetVolume
    threshold = 0.5
    variable = phi
    location = outside
    execute_on = 'initial timestep_end'
  [../]

  [./cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial'
  [../]

[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  end_time = 1
  nl_rel_tol = 1e-12
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  [./TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 1
  [../]

[]

[MultiApps]
  [./reinit]
    type = LevelSetReinitializationMultiApp
    input_files = 'reinit.i'
    execute_on = 'timestep_end'
  [../]
[]

[Transfers]
  [./to_sub]
    type = MultiAppCopyTransfer
    variable = phi
    source_variable = phi
    direction = to_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  [../]

  [./to_sub_init]
    type = MultiAppCopyTransfer
    variable = phi_0
    source_variable = phi
    direction = to_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  [../]

  [./from_sub]
    type = MultiAppCopyTransfer
    variable = phi
    source_variable = phi
    direction = from_multiapp
    multi_app = reinit
    execute_on = timestep_end
  [../]
[]

[Outputs]
  exodus = true
  csv = true
[]

modules/level_set/examples/rotating_circle/circle_rotate_master.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  nx = 32
  ny = 32
  uniform_refine = 2
[]

[AuxVariables]
  [./vel_x]
  [../]
  [./vel_y]
  [../]
[]

[AuxKernels]
  [./vel_x]
    type = FunctionAux
    function = 4*y
    variable = vel_x
    execute_on = initial
  [../]
  [./vel_y]
    type = FunctionAux
    function = -4*x
    variable = vel_y
    execute_on = initial
  [../]
[]

[Variables]
  [./phi]
  [../]
[]

[BCs]
  [./all]
    type = DirichletBC
    variable = phi
    boundary = 'top bottom left right'
    value = 0
  [../]
[]

[Functions]
  [./phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.03
    center = '0 0.5 0'
    radius = 0.15
  [../]
[]

[ICs]
  [./phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]

  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

[Postprocessors]
  [./area]
    type = LevelSetVolume
    threshold = 0.5
    variable = phi
    location = outside
    execute_on = 'initial timestep_end'
  [../]
  [./cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial timestep_end'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  end_time = 1.570796
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  [./TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 0.8
  [../]
[]

[MultiApps]
  [./reinit]
    type = LevelSetReinitializationMultiApp
    input_files = 'circle_rotate_sub.i'
    execute_on = 'timestep_end'
  [../]
[]

[Transfers]
  [./to_sub]
    type = MultiAppCopyTransfer
    source_variable = phi
    variable = phi
    direction = to_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  [../]
  [./to_sub_init]
    type = MultiAppCopyTransfer
    source_variable = phi
    variable = phi_0
    direction = to_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  [../]
  [./from_sub]
    type = MultiAppCopyTransfer
    source_variable = phi
    variable = phi
    direction = from_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  [../]
[]

[Outputs]
  csv = true
  exodus = true
[]

modules/level_set/examples/rotating_circle/circle_rotate_supg.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  nx = 32
  ny = 32
  uniform_refine = 2
[]

[AuxVariables]
  [./vel_x]
  [../]
  [./vel_y]
  [../]
[]

[AuxKernels]
  [./vel_x]
    type = FunctionAux
    function = 4*y
    variable = vel_x
    execute_on = initial
  [../]
  [./vel_y]
    type = FunctionAux
    function = -4*x
    variable = vel_y
    execute_on = initial
  [../]
[]

[Variables]
  [./phi]
  [../]
[]

[Functions]
  [./phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.03
    center = '0 0.5 0'
    radius = 0.15
  [../]
[]

[ICs]
  [./phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]
  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
  [./advection_supg]
    type = LevelSetAdvectionSUPG
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
  [./time_supg]
    type = LevelSetTimeDerivativeSUPG
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

[Postprocessors]
  [./area]
    type = LevelSetVolume
    threshold = 0.5
    variable = phi
    location = outside
    execute_on = 'initial timestep_end'
  [../]
  [./cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  end_time = 1.570796
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  [./TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 0.8
  [../]
[]

[Outputs]
  csv = true
  exodus = true
[]

modules/level_set/examples/vortex/vortex_reinit.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmax = 1
  ymax = 1
  nx = 16
  ny = 16
  uniform_refine = 2
  elem_type = QUAD9
  second_order = true
[]

[AuxVariables]
  [vel_x]
    family = LAGRANGE
  []
  [vel_y]
    family = LAGRANGE
  []
[]

[AuxKernels]
  [vel_x]
    type = FunctionAux
    function = vel_x
    variable = vel_x
    execute_on = 'initial timestep_begin'
  []
  [vel_y]
    type = FunctionAux
    function = vel_y
    variable = vel_y
    execute_on = 'initial timestep_begin'
  []
[]

[Variables]
  [phi]
    family = LAGRANGE
  []
[]

[Functions]
  [phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.03
    center = '0.5 0.75 0'
    radius = 0.15
  []
  [vel_x]
    type = LevelSetOlssonVortex
    component = x
    reverse_time = 2
  []
  [vel_y]
    type = LevelSetOlssonVortex
    component = y
    reverse_time = 2
  []
[]

[ICs]
  [phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  []
[]

[Kernels]
  [time]
    type = TimeDerivative
    variable = phi
  []
  [advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  []
  [advection_supg]
    type = LevelSetAdvectionSUPG
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  []
  [time_supg]
    type = LevelSetTimeDerivativeSUPG
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  []
[]

[Postprocessors]
  [area]
    type = LevelSetVolume
    threshold = 0.5
    variable = phi
    location = outside
    execute_on = 'initial timestep_end'
  []
  [cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial timestep_end'
  []
[]

[Problem]
  type = LevelSetProblem
[]

[Preconditioning/smp]
    type = SMP
    full = true
[]

[Executioner]
  type = Transient
  solve_type = NEWTON
  start_time = 0
  end_time = 2
  scheme = crank-nicolson
  [TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 0.8
  []
[]

[MultiApps]
  [reinit]
    type = LevelSetReinitializationMultiApp
    input_files = 'vortex_reinit_sub.i'
    execute_on = TIMESTEP_END
  []
[]

[Transfers]
  [to_sub]
    type = MultiAppCopyTransfer
    source_variable = phi
    variable = phi
    direction = to_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  []

  [to_sub_init]
    type = MultiAppCopyTransfer
    source_variable = phi
    variable = phi_0
    direction = to_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  []

  [from_sub]
    type = MultiAppCopyTransfer
    source_variable = phi
    variable = phi
    direction = from_multiapp
    multi_app = reinit
    execute_on = 'timestep_end'
  []
[]

[Outputs]
  csv = true
  exodus = true
[]

modules/level_set/examples/vortex/vortex_supg.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmax = 1
  ymax = 1
  nx = 16
  ny = 16
  uniform_refine = 2
  elem_type = QUAD9
[]

[AuxVariables]
  [./vel_x]
    family = LAGRANGE
    order = FIRST
  [../]
  [./vel_y]
    family = LAGRANGE
    order = FIRST
  [../]
[]

[AuxKernels]
  [./vel_x]
    type = FunctionAux
    function = vel_x
    variable = vel_x
    execute_on = 'initial timestep_begin'
  [../]
  [./vel_y]
    type = FunctionAux
    function = vel_y
    variable = vel_y
    execute_on = 'initial timestep_begin'
  [../]
[]

[Variables]
  [./phi]
    family = LAGRANGE
    order = FIRST
  [../]
[]

[Functions]
  [./phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.01184
    center = '0.5 0.75 0'
    radius = 0.15
  [../]
  [./vel_x]
    type = LevelSetOlssonVortex
    component = x
    reverse_time = 2
  [../]
  [./vel_y]
    type = LevelSetOlssonVortex
    component = y
    reverse_time = 2
  [../]
[]

[ICs]
  [./phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]
  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
  [./advection_supg]
    type = LevelSetAdvectionSUPG
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
  [./time_supg]
    type = LevelSetTimeDerivativeSUPG
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

[Postprocessors]
  [./area]
    type = LevelSetVolume
    threshold = 0.5
    variable = phi
    location = outside
    execute_on = 'initial timestep_end'
  [../]
  [./cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial timestep_end'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  end_time = 2
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  [./TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 0.8
  [../]
[]

[Outputs]
  csv = true
  exodus = true
[]

modules/level_set/examples/rotating_circle/circle_rotate.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  nx = 32
  ny = 32
  uniform_refine = 2
[]

[AuxVariables]
  [./vel_x]
  [../]
  [./vel_y]
  [../]
[]

[AuxKernels]
  [./vel_x]
    type = FunctionAux
    function = 4*y
    variable = vel_x
    execute_on = initial
  [../]
  [./vel_y]
    type = FunctionAux
    function = -4*x
    variable = vel_y
    execute_on = initial
  [../]
[]

[Variables]
  [./phi]
  [../]
[]

[Functions]
  [./phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.03
    center = '0 0.5 0'
    radius = 0.15
  [../]
[]

[ICs]
  [./phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]

  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

[Postprocessors]
  [./area]
    type = LevelSetVolume
    threshold = 0.5
    variable = phi
    location = outside
    execute_on = 'initial timestep_end'
  [../]
  [./cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial' #timestep_end'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  end_time = 1.570796
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  [./TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 0.8
  [../]
[]

[Outputs]
  csv = true
  exodus = true
[]

modules/level_set/test/tests/verification/1d_level_set_supg_mms/1d_level_set_supg_mms.i

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = 0
  xmax = 32
  nx = 64
[]

[Variables]
  [./phi]
  [../]
[]

[AuxVariables]
  [./v_x]
    initial_condition = 1
  [../]
[]

[ICs]
  [./phi_ic]
    function = phi_exact
    variable = phi
    type = FunctionIC
  [../]
[]

[Functions]
  [./phi_exact]
    type = ParsedFunction
    value = 'a*exp(1/(10*t))*sin(2*pi*x/b) + 1'
    vars = 'a b'
    vals = '1 8'
  [../]
  [./phi_mms]
    type = ParsedFunction
    value = '-a*exp(1/(10*t))*sin(2*pi*x/b)/(10*t^2) + 2*pi*a*exp(1/(10*t))*cos(2*pi*x/b)/b'
    vars = 'a b'
    vals = '1 8'
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]
  [./time_supg]
    type = LevelSetTimeDerivativeSUPG
    variable = phi
    velocity_x = v_x
  [../]
  [./phi_advection]
    type = LevelSetAdvection
    variable = phi
    velocity_x = v_x
  [../]
  [./phi_forcing]
    type = BodyForce
    variable = phi
    function = phi_mms
  [../]
  [./phi_advection_supg]
    type = LevelSetAdvectionSUPG
    variable = phi
    velocity_x = v_x
  [../]
  [./phi_forcing_supg]
    type = LevelSetForcingFunctionSUPG
    velocity_x = v_x
    variable = phi
    function = phi_mms
  [../]
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = phi_exact
    variable = phi
  [../]
  [./h]
    type = AverageElementSize
  [../]
  [./point]
    type = PointValue
    point = '0.1 0 0'
    variable = phi
  [../]
[]

[Executioner]
  type = Transient
  start_time = 1
  dt = 0.01
  end_time = 1.25
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  scheme = bdf2
  nl_rel_tol = 1e-12
[]

[Outputs]
  interval = 10
  execute_on = 'timestep_end'
  csv = true
[]

modules/level_set/test/tests/kernels/advection/advection_mms.i

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = 0
  xmax = 12
  nx = 48
[]

[Adaptivity]
  steps = 5
  marker = marker
  [./Markers]
    [./marker]
      type = UniformMarker
      mark = REFINE
    [../]
  [../]
[]

[Variables]
  [./phi]
  [../]
[]

[AuxVariables]
  [./v_x]
    initial_condition = 2
  [../]
[]

[BCs]
  [./left]
    type = FunctionDirichletBC
    boundary = 'left'
    function = phi_exact
    variable = phi
  [../]
[]

[Functions]
  [./phi_exact]
    type = ParsedFunction
    value = 'a*sin(pi*x/b)*cos(pi*x)'
    vars = 'a b'
    vals = '2 12'
  [../]
  [./phi_mms]
    type = ParsedFunction
    value = '-2*pi*a*sin(pi*x)*sin(pi*x/b) + 2*pi*a*cos(pi*x)*cos(pi*x/b)/b'
    vars = 'a b'
    vals = '2 12'
  [../]
[]

[Kernels]
  [./phi_advection]
    type = LevelSetAdvection
    variable = phi
    velocity_x = v_x
  [../]
  [./phi_forcing]
    type = BodyForce
    variable = phi
    function = phi_mms
  [../]
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = phi_exact
    variable = phi
  [../]
  [./h]
    type = AverageElementSize
  [../]
[]

[VectorPostprocessors]
  active = ''
  [./results]
    type = LineValueSampler
    variable = phi
    start_point = '0 0 0'
    end_point = '12 0 0'
    num_points = 500
    sort_by = x
  [../]
[]

[Executioner]
  type = Steady
  nl_rel_tol = 1e-10
  solve_type = NEWTON
  # A steady-state pure advection problem is numerically challenging,
  # it has a zero diagonal in the Jabocian matrix. The following solver
  # settings seem to reliably solve this problem.
  petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
  petsc_options_value = 'lu       superlu_dist'
[]

[Outputs]
  execute_on = 'TIMESTEP_END'
  csv = true
[]

modules/level_set/examples/circle/circle_16.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
  nx = 16
  ny = 16
[]

[Variables]
  [./phi]
  [../]
[]

[AuxVariables]
  [./vel_x]
    initial_condition = 3
  [../]
  [./vel_y]
    initial_condition = 3
  [../]
[]

[ICs]
  [./phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  [../]
[]

[Functions]
  [./phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.05
    center = '0.5 0.5 0'
    radius = 0.15
  [../]
[]

[BCs]
  [./Periodic]
    [./all]
      variable = phi
      auto_direction = 'x y'
    [../]
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]

  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

[Postprocessors]
  [./cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  end_time = 1
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  [./TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 0.8
  [../]
[]

[Outputs]
  csv = true
  exodus = true
[]

modules/level_set/test/tests/verification/1d_level_set_mms/level_set_mms.i

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = 0
  xmax = 32
  nx = 64
  uniform_refine = 0
[]

[Variables]
  [./phi]
  [../]
[]

[AuxVariables]
  [./v_x]
    initial_condition = 1
  [../]
[../]

[ICs]
  [./phi_ic]
    function = phi_exact
    variable = phi
    type = FunctionIC
  [../]
[]

[Functions]
  [./phi_exact]
    type = ParsedFunction
    value = 'a*exp(1/(10*t))*sin(2*pi*x/b) + 1'
    vars = 'a b'
    vals = '1 8'
  [../]
  [./phi_mms]
    type = ParsedFunction
    value = '-a*exp(1/(10*t))*sin(2*pi*x/b)/(10*t^2) + 2*pi*a*exp(1/(10*t))*cos(2*pi*x/b)/b'
    vars = 'a b'
    vals = '1 8'
  [../]
[]

[Kernels]
  [./phi_advection]
    type = LevelSetAdvection
    variable = phi
    velocity_x = v_x
  [../]
  [./phi_time]
    type = TimeDerivative
    variable = phi
  [../]
  [./phi_forcing]
    type = BodyForce
    variable = phi
    function = phi_mms
  [../]
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    function = phi_exact
    variable = phi
  [../]
  [./h]
    type = AverageElementSize
  [../]
  [./point]
    type = PointValue
    point = '0.1 0 0'
    variable = phi
  [../]
[]

[Executioner]
  type = Transient
  start_time = 1
  dt = 0.01
  end_time = 1.25
  solve_type = PJFNK
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  scheme = bdf2
  nl_rel_tol = 1e-12
[]

[Outputs]
  interval = 10
  execute_on = 'timestep_end'
  csv = true
[]

modules/level_set/examples/vortex/vortex.i

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmax = 1
  ymax = 1
  nx = 16
  ny = 16
  uniform_refine = 2
  elem_type = QUAD9
[]

[AuxVariables]
  [./vel_x]
    family = LAGRANGE
    order = FIRST
  [../]
  [./vel_y]
    family = LAGRANGE
    order = FIRST
  [../]
[]

[AuxKernels]
  [./vel_x]
    type = FunctionAux
    function = vel_x
    variable = vel_x
    execute_on = 'initial timestep_begin'
  [../]
  [./vel_y]
    type = FunctionAux
    function = vel_y
    variable = vel_y
    execute_on = 'initial timestep_begin'
  [../]
[]

[Variables]
  [./phi]
    family = LAGRANGE
    order = FIRST
  [../]
[]

[Functions]
  [./phi_exact]
    type = LevelSetOlssonBubble
    epsilon = 0.01184
    center = '0.5 0.75 0'
    radius = 0.15
  [../]
  [./vel_x]
    type = LevelSetOlssonVortex
    component = x
    reverse_time = 2
  [../]
  [./vel_y]
    type = LevelSetOlssonVortex
    component = y
    reverse_time = 2
  [../]
[]

[ICs]
  [./phi_ic]
    type = FunctionIC
    function = phi_exact
    variable = phi
  [../]
[]

[Kernels]
  [./time]
    type = TimeDerivative
    variable = phi
  [../]
  [./advection]
    type = LevelSetAdvection
    velocity_x = vel_x
    velocity_y = vel_y
    variable = phi
  [../]
[]

[Postprocessors]
  [./area]
    type = LevelSetVolume
    threshold = 0.5
    variable = phi
    location = outside
    execute_on = 'initial timestep_end'
  [../]
  [./cfl]
    type = LevelSetCFLCondition
    velocity_x = vel_x
    velocity_y = vel_y
    execute_on = 'initial timestep_end'
  [../]
[]

[Executioner]
  type = Transient
  solve_type = PJFNK
  start_time = 0
  end_time = 2
  scheme = crank-nicolson
  petsc_options_iname = '-pc_type -pc_sub_type'
  petsc_options_value = 'asm      ilu'
  [./TimeStepper]
    type = PostprocessorDT
    postprocessor = cfl
    scale = 0.8
  [../]
[]

[Outputs]
  csv = true
  exodus = true
[]

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