FVAdvection

Residual contribution from advection operator for finite volume method.

The FVAdvection kernel implements an advection term given for the domain ( $var element = document.getElementById("moose-equation-dfd034f2-85c6-4810-a3d3-e13f8c775fb6");katex.render("\\Omega", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ) defined as

var element = document.getElementById("moose-equation-8e0877fd-b916-4b7f-8250-8157f92ed661");katex.render("\\underbrace{\\vec{v} \\cdot \\nabla u} + \\sum_{i=1}^n \\beta_i = 0 \\in \\Omega,", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

where $var element = document.getElementById("moose-equation-d17b60a0-52a6-4d4f-9d47-28d2d6bc2d9c");katex.render("u", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the advected quantity, the variable for this kernel, $var element = document.getElementById("moose-equation-00e57830-e3b7-4903-a77d-ba0922ee409e");katex.render("v", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the constant advecting velocity, the velocity parameter of this kernel, and the $var element = document.getElementById("moose-equation-3c96df84-cb38-4e25-9ba4-7b48d7465e1c");katex.render("\\beta_i", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ are the contribution to the residual of other kernels.

This volumetric term is transformed using the divergence theorem into a surface integral, computed as a sum over each face of the advective flux. This is preferred over computing a volumetric gradient as conservative advection is naturally achieved.

var element = document.getElementById("moose-equation-1fc601ca-1db3-4940-8a56-f7517e94e57b");katex.render("\\int_{element} \\vec{v} \\cdot \\nabla u = \\sum_{elem faces f} u_f \\vec{v} \\cdot n_f v area_f", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

The advected quantity is evaluated on the face using an advected_interp(olation)_method. Two methods are available:

average for a geometrically weighted average between the element and neighbor values
upwind for a first order upwind scheme, which uses the value from the centroid of the element situated upwind of the face, using velocity as the wind

note

This kernel leverages the automatic differentiation system, so the Jacobian is computed at the same time as the residual and need not be defined separately.

Boundary conditions for pure advection

Advection problems, with a constant advecting velocity, should have two types of boundary conditions: inflow and outflow. The inflow boundary conditions may be specified as a constant boundary value with a FVDirichletBC (with caveats, see documentation) or as a constant flux with FVAdvectionFunctionBC.

[FVBCs]
  [advection]
    type = FVAdvectionFunctionBC
    boundary = 'left right'
    exact_solution = 'exact'
    variable = v
    velocity = '${a} 0 0'
  []
[]

The outflow boundary conditions may be specified with a FVConstantScalarOutflowBC.

[FVBCs]
  [fv_outflow]
    type = FVConstantScalarOutflowBC
    velocity = '1 0.5 0'
    variable = v
    boundary = 'right top'
  []
[]

If no boundary conditions are specified, then there is a zero advective flux through the boundary, also known a no-penetration boundary condition.

note

The FVAdvection kernel may be executed on boundaries using the force_boundary_execution and boundaries_to_force parameters, however this is somewhat situational / not for mainstream use.

Example input syntax

In this example, a simple time-dependent advection problem is solved, with a constant advecting velocity of 1 0.5 0.

[FVKernels]
  [advection]
    type = FVAdvection
    variable = v
    velocity = '1 0.5 0'
  []
  [time]
    type = FVTimeKernel
    variable = v
  []
[]

Input Parameters

variableThe name of the finite volume variable this kernel applies to
C++ Type:NonlinearVariableName
Controllable:No
Description:The name of the finite volume variable this kernel applies to
velocityConstant advection velocity
C++ Type:libMesh::VectorValue<double>
Controllable:No
Description:Constant advection velocity

Required Parameters

advected_interp_methodupwindThe interpolation to use for the advected quantity. Options are 'upwind', 'average' and 'skewness-corrected', with the default being 'upwind'.
Default:upwind
C++ Type:MooseEnum
Options:average, upwind, skewness-corrected
Controllable:No
Description:The interpolation to use for the advected quantity. Options are 'upwind', 'average' and 'skewness-corrected', with the default being 'upwind'.
blockThe list of blocks (ids or names) that this object will be applied
C++ Type:std::vector<SubdomainName>
Controllable:No
Description:The list of blocks (ids or names) that this object will be applied
boundaries_to_forceThe set of boundaries to force execution of this FVFluxKernel on.
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The set of boundaries to force execution of this FVFluxKernel on.
boundaries_to_not_forceThe set of boundaries to not force execution of this FVFluxKernel on.
C++ Type:std::vector<BoundaryName>
Controllable:No
Description:The set of boundaries to not force execution of this FVFluxKernel on.
force_boundary_executionFalseWhether to force execution of this object on the boundary.
Default:False
C++ Type:bool
Controllable:No
Description:Whether to force execution of this object on the boundary.
ghost_layers1The number of layers of elements to ghost.
Default:1
C++ Type:unsigned short
Controllable:No
Description:The number of layers of elements to ghost.
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
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_point_neighborsFalseWhether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.
Default:False
C++ Type:bool
Controllable:No
Description:Whether to use point neighbors, which introduces additional ghosting to that used for simple face neighbors.

Optional 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.
implicitTrueDetermines whether this object is calculated using an implicit or explicit form
Default:True
C++ Type:bool
Controllable:No
Description:Determines whether this object is calculated using an implicit or explicit form
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

extra_matrix_tagsThe extra tags for the matrices this Kernel should fill
C++ Type:std::vector<TagName>
Controllable:No
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<TagName>
Controllable:No
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
Controllable:No
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
Controllable:No
Description:The tag for the vectors this Kernel should fill

Tagging Parameters

Input Files

(test/tests/indicators/analytical_indicator/analytical_indicator_fv.i)
(test/tests/fvkernels/mms/advective-outflow/advection-diffusion.i)
(test/tests/fvkernels/mms/cylindrical/advection-reaction.i)
(test/tests/fvkernels/mms/non-orthogonal/extended-adr.i)
(test/tests/fvkernels/mms/advective-outflow/advection.i)
(test/tests/fvkernels/mms/advective-outflow/advection-outflow.i)
(test/tests/fvkernels/mms/advection.i)
(test/tests/fvkernels/mms/skewness-correction/two_term_extrapol/advection-outflow.i)
(test/tests/markers/error_fraction_marker/error_fraction_marker_fv.i)
(test/tests/fvkernels/mms/non-orthogonal/advection-diffusion-reaction.i)
(test/tests/fvkernels/fv_constant_scalar_advection/2D_constant_scalar_advection.i)
(test/tests/fvkernels/mms/skewness-correction/adv-diff-react/skewed.i)
(test/tests/fvkernels/mms/advection-diffusion.i)
(test/tests/fvkernels/mms/cylindrical/advection-diffusion-reaction.i)
(test/tests/fvkernels/mms/cylindrical/advection.i)

(test/tests/fvkernels/mms/advection.i)

a=1.1

[GlobalParams]
  advected_interp_method = 'average'
[]

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = -0.6
    xmax = 0.6
    nx = 2
  [../]
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
  [../]
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [advection]
    type = FVAdvectionFunctionBC
    boundary = 'left right'
    exact_solution = 'exact'
    variable = v
    velocity = '${a} 0 0'
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = '1.1 * sin(1.1 * x)'
  []
  [forcing]
    type = ParsedFunction
    value = '${a} * 1.1 * 1.1 * cos(1.1 * x)'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      NONZERO'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/fv_constant_scalar_advection/2D_constant_scalar_advection.i)

[Mesh]
  [gen_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 2
    ymin = 0
    ymax = 4
    nx = 10
    ny = 20
  []
[]

[Variables]
  [v]
    type = MooseVariableFVReal
    two_term_boundary_expansion = false
  []
[]

[ICs]
  [v_ic]
    type = FunctionIC
    variable = v
    function = 'r2 := (x - 0.5)*(x - 0.5) + (y - 0.3)*(y - 0.3); exp(-r2 * 20)'
  []
[]

[FVKernels]
  [advection]
    type = FVAdvection
    variable = v
    velocity = '1 0.5 0'
  []
  [time]
    type = FVTimeKernel
    variable = v
  []
[]

[FVBCs]
  [fv_outflow]
    type = FVConstantScalarOutflowBC
    velocity = '1 0.5 0'
    variable = v
    boundary = 'right top'
  []
[]

[Executioner]
  type = Transient
  petsc_options = '-snes_converged_reason'
  num_steps = 10
  dt = 0.25
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
[]

(test/tests/fvkernels/fv_constant_scalar_advection/2D_constant_scalar_advection.i)

[Mesh]
  [gen_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 2
    ymin = 0
    ymax = 4
    nx = 10
    ny = 20
  []
[]

[Variables]
  [v]
    type = MooseVariableFVReal
    two_term_boundary_expansion = false
  []
[]

[ICs]
  [v_ic]
    type = FunctionIC
    variable = v
    function = 'r2 := (x - 0.5)*(x - 0.5) + (y - 0.3)*(y - 0.3); exp(-r2 * 20)'
  []
[]

[FVKernels]
  [advection]
    type = FVAdvection
    variable = v
    velocity = '1 0.5 0'
  []
  [time]
    type = FVTimeKernel
    variable = v
  []
[]

[FVBCs]
  [fv_outflow]
    type = FVConstantScalarOutflowBC
    velocity = '1 0.5 0'
    variable = v
    boundary = 'right top'
  []
[]

[Executioner]
  type = Transient
  petsc_options = '-snes_converged_reason'
  num_steps = 10
  dt = 0.25
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
[]

(test/tests/indicators/analytical_indicator/analytical_indicator_fv.i)

[Mesh]
  [mesh]
  type = GeneratedMeshGenerator
  dim = 2
  nx = 20
  ny = 1
  []
[]

[Variables]
  [u]
    order = CONSTANT
    family = MONOMIAL
    fv = true
  []
[]

[Functions]
  [solution]
    type = ParsedFunction
    value = (exp(x)-1)/(exp(1)-1)
  []
[]

[FVKernels]
  [diff]
    type = FVDiffusion
    variable = u
    coeff = coeff
  []
  [conv]
    type = FVAdvection
    variable = u
    velocity = '1 0 0'
  []
[]

[FVBCs]
  [left]
    type = FVDirichletBC
    variable = u
    boundary = left
    value = 0
  []
  [right]
    type = FVDirichletBC
    variable = u
    boundary = right
    value = 1
  []
[]

[Materials]
  [diff]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff'
    prop_values = '1'
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
[]

[Adaptivity]
  [Indicators]
    [error]
      type = AnalyticalIndicator
      variable = u
      function = solution
    []
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/fvkernels/mms/advective-outflow/advection-diffusion.i)

diff=1
a=1

[GlobalParams]
  advected_interp_method = 'average'
[]

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = -1
    xmax = 0
    nx = 2
  [../]
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
    force_boundary_execution = true
  [../]
  [./diffusion]
    type = FVDiffusion
    variable = v
    coeff = coeff
  [../]
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [left]
    type = FVFunctionDirichletBC
    boundary = 'left'
    function = 'exact'
    variable = v
  []
[]

[Materials]
  [diff]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff'
    prop_values = '${diff}'
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = 'cos(x)'
  []
  [forcing]
    type = ParsedFunction
    value = 'cos(x) - sin(x)'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/mms/cylindrical/advection-reaction.i)

a=1.1

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 2
    xmax = 3
    ymin = 0
    ymax = 1
    nx = 2
    ny = 2
  [../]
[]

[Problem]
  coord_type = 'RZ'
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    initial_condition = 1
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} ${a} 0'
    advected_interp_method = 'average'
  [../]
  [reaction]
    type = FVReaction
    variable = v
  []
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [advection]
    type = FVAdvectionFunctionBC
    boundary = 'left right top bottom'
    exact_solution = 'exact'
    variable = v
    velocity = '${a} ${a} 0'
    advected_interp_method = 'average'
  []
[]

[Functions]
[exact]
  type = ParsedFunction
  value = 'sin(x)*cos(y)'
[]
[forcing]
  type = ParsedFunction
  value = '-a*sin(x)*sin(y) + sin(x)*cos(y) + (x*a*cos(x)*cos(y) + a*sin(x)*cos(y))/x'
  vars = 'a'
  vals = '${a}'
[]
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -sub_pc_factor_shift_type -sub_pc_type'
  petsc_options_value = 'asm      NONZERO                   lu'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/mms/non-orthogonal/extended-adr.i)

a=1.1
diff=1.1

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 2
    xmax = 3
    ymin = 0
    ymax = 1
    nx = 2
    ny = 2
    elem_type = TRI3
  [../]
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    initial_condition = 1
    type = MooseVariableFVReal
    face_interp_method = vertex-based
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} ${fparse 2*a} 0'
    advected_interp_method = 'average'
  [../]
  [reaction]
    type = FVReaction
    variable = v
  []
  [diff_v]
    type = FVDiffusion
    variable = v
    coeff = ${diff}
    use_point_neighbors = true
  []
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [exact]
    type = FVFunctionDirichletBC
    boundary = 'left right top bottom'
    function = 'exact'
    variable = v
  []
[]

[Functions]
[exact]
  type = ParsedFunction
  value = 'sin(x)*cos(y)'
[]
[forcing]
  type = ParsedFunction
  value = '-2*a*sin(x)*sin(y) + a*cos(x)*cos(y) + 2*diff*sin(x)*cos(y) + sin(x)*cos(y)'
  vars = 'a diff'
  vals = '${a} ${diff}'
[]
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'hypre'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/mms/advective-outflow/advection.i)

a=1.1

[GlobalParams]
  advected_interp_method = 'average'
[]

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 0.1
    xmax = 1.1
    nx = 2
  [../]
[]

[Variables]
  [./u]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    two_term_boundary_expansion = false
    type = MooseVariableFVReal
  [../]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    two_term_boundary_expansion = true
    type = MooseVariableFVReal
  [../]
[]

[FVKernels]
  [./advection_u]
    type = FVAdvection
    variable = u
    velocity = '${a} 0 0'
    force_boundary_execution = true
  [../]
  [body_u]
    type = FVBodyForce
    variable = u
    function = 'forcing'
  []
  [./advection_v]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
    force_boundary_execution = true
  [../]
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [left_u]
    type = FVFunctionDirichletBC
    boundary = 'left'
    function = 'exact'
    variable = u
  []
  [left_v]
    type = FVFunctionDirichletBC
    boundary = 'left'
    function = 'exact'
    variable = v
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = 'cos(x)'
  []
  [forcing]
    type = ParsedFunction
    value = '-${a} * sin(x)'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./L2u]
    type = ElementL2Error
    variable = u
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [./L2v]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/mms/advective-outflow/advection-outflow.i)

a=1.1

[GlobalParams]
  advected_interp_method = 'average'
[]

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 0
    xmax = 1.1
    nx = 2
  [../]
[]

[Variables]
  [./u]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    two_term_boundary_expansion = false
    type = MooseVariableFVReal
  [../]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    two_term_boundary_expansion = true
    type = MooseVariableFVReal
  [../]
[]

[FVKernels]
  [./advection_u]
    type = FVAdvection
    variable = u
    velocity = '${a} 0 0'
  [../]
  [body_u]
    type = FVBodyForce
    variable = u
    function = 'forcing'
  []
  [./advection_v]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
  [../]
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [left_u]
    type = FVFunctionDirichletBC
    boundary = 'left'
    function = 'exact'
    variable = u
  []
  [right_u]
    type = FVConstantScalarOutflowBC
    variable = u
    velocity = '${a} 0 0'
    boundary = 'right'
  []
  [left_v]
    type = FVFunctionDirichletBC
    boundary = 'left'
    function = 'exact'
    variable = v
  []
  [right_v]
    type = FVConstantScalarOutflowBC
    variable = v
    velocity = '${a} 0 0'
    boundary = 'right'
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = 'cos(x)'
  []
  [forcing]
    type = ParsedFunction
    value = '-${a} * sin(x)'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./L2u]
    type = ElementL2Error
    variable = u
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [./L2v]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/mms/advection.i)

a=1.1

[GlobalParams]
  advected_interp_method = 'average'
[]

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = -0.6
    xmax = 0.6
    nx = 2
  [../]
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
  [../]
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [advection]
    type = FVAdvectionFunctionBC
    boundary = 'left right'
    exact_solution = 'exact'
    variable = v
    velocity = '${a} 0 0'
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = '1.1 * sin(1.1 * x)'
  []
  [forcing]
    type = ParsedFunction
    value = '${a} * 1.1 * 1.1 * cos(1.1 * x)'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -sub_pc_factor_shift_type'
  petsc_options_value = 'asm      NONZERO'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/mms/skewness-correction/two_term_extrapol/advection-outflow.i)

diff=1
a=1

[GlobalParams]
  advected_interp_method = 'average'
[]

[Mesh]
  [./gen_mesh]
    type = FileMeshGenerator
    file = skewed.msh
  [../]
[]

[Variables]
  [./v]
    type = MooseVariableFVReal
    face_interp_method = 'skewness-corrected'
    cache_face_gradients = false
    cache_face_values = true
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
  [../]
  [./diffusion]
    type = FVDiffusion
    variable = v
    coeff = coeff
  [../]
  [./body]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  [../]
[]

[FVBCs]
  [left]
    type = FVFunctionDirichletBC
    boundary = 'left'
    function = 'exact'
    variable = v
  []
  [top]
    type = FVNeumannBC
    boundary = 'top'
    value = 0
    variable = v
  []
  [bottom]
    type = FVNeumannBC
    boundary = 'bottom'
    value = 0
    variable = v
  []
  [right]
    type = FVConstantScalarOutflowBC
    variable = v
    velocity = '${a} 0 0'
    boundary = 'right'
  []
[]

[Materials]
  [diff]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff'
    prop_values = '${diff}'
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = 'cos(x)'
  []
  [forcing]
    type = ParsedFunction
    value = 'cos(x) - sin(x)'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_hypre_type -snes_linesearch_minlambda'
  petsc_options_value = 'hypre boomeramg 1e-9'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/markers/error_fraction_marker/error_fraction_marker_fv.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  nx = 10
  ny = 10
[]

[Variables]
  [u]
    order = CONSTANT
    family = MONOMIAL
    fv = true
  []
[]

[Functions]
  [solution]
    type = ParsedFunction
    value = (exp(x)-1)/(exp(1)-1)
  []
[]

[FVKernels]
  [diff]
    type = FVDiffusion
    variable = u
    coeff = coeff
  []
  [conv]
    type = FVAdvection
    variable = u
    velocity = '1 0 0'
  []
[]

[FVBCs]
  [left]
    type = FVDirichletBC
    variable = u
    boundary = left
    value = 0
  []
  [right]
    type = FVDirichletBC
    variable = u
    boundary = right
    value = 1
  []
[]

[Materials]
  [diff]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff'
    prop_values = '1'
  []
[]

[Executioner]
  type = Steady
  solve_type = NEWTON
[]

[Adaptivity]
  [Indicators]
    [error]
      type = AnalyticalIndicator
      variable = u
      function = solution
    []
  []
  [Markers]
    [marker]
      type = ErrorFractionMarker
      coarsen = 0.1
      indicator = error
      refine = 0.3
    []
  []
[]

[Outputs]
  exodus = true
[]

(test/tests/fvkernels/mms/non-orthogonal/advection-diffusion-reaction.i)

a=1.1
diff=1.1

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 2
    xmax = 3
    ymin = 0
    ymax = 1
    nx = 2
    ny = 2
    elem_type = TRI3
  [../]
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    initial_condition = 1
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} ${fparse 2*a} 0'
    advected_interp_method = 'average'
  [../]
  [reaction]
    type = FVReaction
    variable = v
  []
  [diff_v]
    type = FVDiffusion
    variable = v
    coeff = ${diff}
  []
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [exact]
    type = FVFunctionDirichletBC
    boundary = 'left right top bottom'
    function = 'exact'
    variable = v
  []
[]

[Functions]
[exact]
  type = ParsedFunction
  value = 'sin(x)*cos(y)'
[]
[forcing]
  type = ParsedFunction
  value = '-2*a*sin(x)*sin(y) + a*cos(x)*cos(y) + 2*diff*sin(x)*cos(y) + sin(x)*cos(y)'
  vars = 'a diff'
  vals = '${a} ${diff}'
[]
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'hypre'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/fv_constant_scalar_advection/2D_constant_scalar_advection.i)

[Mesh]
  [gen_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 0
    xmax = 2
    ymin = 0
    ymax = 4
    nx = 10
    ny = 20
  []
[]

[Variables]
  [v]
    type = MooseVariableFVReal
    two_term_boundary_expansion = false
  []
[]

[ICs]
  [v_ic]
    type = FunctionIC
    variable = v
    function = 'r2 := (x - 0.5)*(x - 0.5) + (y - 0.3)*(y - 0.3); exp(-r2 * 20)'
  []
[]

[FVKernels]
  [advection]
    type = FVAdvection
    variable = v
    velocity = '1 0.5 0'
  []
  [time]
    type = FVTimeKernel
    variable = v
  []
[]

[FVBCs]
  [fv_outflow]
    type = FVConstantScalarOutflowBC
    velocity = '1 0.5 0'
    variable = v
    boundary = 'right top'
  []
[]

[Executioner]
  type = Transient
  petsc_options = '-snes_converged_reason'
  num_steps = 10
  dt = 0.25
  nl_rel_tol = 1e-12
[]

[Outputs]
  exodus = true
[]

(test/tests/fvkernels/mms/skewness-correction/adv-diff-react/skewed.i)

a=1.1
diff=1.1

[Mesh]
  [gen_mesh]
    type = FileMeshGenerator
    file = skewed.msh
  []
[]


[Variables]
  [v]
    initial_condition = 1
    type = MooseVariableFVReal
    face_interp_method = 'skewness-corrected'
    cache_face_gradients = false
    cache_face_values = true
  []
[]

[FVKernels]
  [diff_v]
    type = FVDiffusion
    variable = v
    coeff = ${diff}
  []
  [advection]
    type = FVAdvection
    variable = v
    velocity = '${a} ${fparse 2*a} 0'
    advected_interp_method = 'average'
  []
  [reaction]
    type = FVReaction
    variable = v
  []
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [exact]
    type = FVFunctionDirichletBC
    boundary = 'left right top bottom'
    function = 'exact'
    variable = v
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = 'sin(x)*cos(y)'
  []
  [forcing]
    type = ParsedFunction
    value = '-2*a*sin(x)*sin(y) + a*cos(x)*cos(y) + 2*diff*sin(x)*cos(y) + sin(x)*cos(y)'
    vars = 'a diff'
    vals = '${a} ${diff}'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'
[]

[Outputs]
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
  []
[]

(test/tests/fvkernels/mms/advection-diffusion.i)

diff=1.1
a=1.1

[GlobalParams]
  advected_interp_method = 'average'
[]

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = -0.6
    xmax = 0.6
    nx = 64
  [../]
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
  [../]
  [./diffusion]
    type = FVDiffusion
    variable = v
    coeff = coeff
  [../]
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [boundary]
    type = FVFunctionDirichletBC
    boundary = 'left right'
    function = 'exact'
    variable = v
  []
[]

[Materials]
  [diff]
    type = ADGenericFunctorMaterial
    prop_names = 'coeff'
    prop_values = '${diff}'
  []
[]

[Functions]
  [exact]
    type = ParsedFunction
    value = '3*x^2 + 2*x + 1'
  []
  [forcing]
    type = ParsedFunction
    value = '-${diff}*6 + ${a} * (6*x + 2)'
    # value = '-${diff}*6'
  []
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/mms/cylindrical/advection-diffusion-reaction.i)

a=1.1
diff=1.1

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 2
    xmin = 2
    xmax = 3
    ymin = 0
    ymax = 1
    nx = 2
    ny = 2
  [../]
[]

[Problem]
  coord_type = 'RZ'
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    initial_condition = 1
  [../]
[]

[FVKernels]
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} ${a} 0'
    advected_interp_method = 'average'
  [../]
  [reaction]
    type = FVReaction
    variable = v
  []
  [diff_v]
    type = FVDiffusion
    variable = v
    coeff = ${diff}
  []
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [exact]
    type = FVFunctionDirichletBC
    boundary = 'left right top bottom'
    function = 'exact'
    variable = v
  []
[]

[Functions]
[exact]
  type = ParsedFunction
  value = 'sin(x)*cos(y)'
[]
[forcing]
  type = ParsedFunction
  value = '-a*sin(x)*sin(y) + diff*sin(x)*cos(y) + sin(x)*cos(y) + (x*a*cos(x)*cos(y) + a*sin(x)*cos(y))/x - (-x*diff*sin(x)*cos(y) + diff*cos(x)*cos(y))/x'
  vars = 'a diff'
  vals = '${a} ${diff}'
[]
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -sub_pc_factor_shift_type -sub_pc_type'
  petsc_options_value = 'asm      NONZERO                   lu'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

(test/tests/fvkernels/mms/cylindrical/advection.i)

a=1.1

[Mesh]
  [./gen_mesh]
    type = GeneratedMeshGenerator
    dim = 1
    xmin = 2
    xmax = 3
    nx = 2
  [../]
[]

[Problem]
  coord_type = 'RZ'
[]

[Variables]
  [./v]
    family = MONOMIAL
    order = CONSTANT
    fv = true
    initial_condition = 1
  [../]
[]

[FVKernels]
  # Flux kernel
  [./advection]
    type = FVAdvection
    variable = v
    velocity = '${a} 0 0'
    advected_interp_method = 'average'
  [../]
  [body_v]
    type = FVBodyForce
    variable = v
    function = 'forcing'
  []
[]

[FVBCs]
  [advection]
    type = FVAdvectionFunctionBC
    boundary = 'left right'
    exact_solution = 'exact'
    variable = v
    velocity = '${a} 0 0'
    advected_interp_method = 'average'
  []
[]

[Functions]
[exact]
  type = ParsedFunction
  value = 'sin(x)'
[]
[forcing]
  type = ParsedFunction
  value = '(x*a*cos(x) + a*sin(x))/x'
  vars = 'a'
  vals = '${a}'
[]
[]

[Executioner]
  type = Steady
  solve_type = 'NEWTON'
  petsc_options_iname = '-pc_type -sub_pc_factor_shift_type -sub_pc_type'
  petsc_options_value = 'asm      NONZERO                   lu'
[]

[Outputs]
  exodus = true
  csv = true
[]

[Postprocessors]
  [./error]
    type = ElementL2Error
    variable = v
    function = exact
    outputs = 'console csv'
    execute_on = 'timestep_end'
  [../]
  [h]
    type = AverageElementSize
    outputs = 'console csv'
    execute_on = 'timestep_end'
  []
[]

Boundary conditions for pure advection
Example input syntax
Input Parameters
Input Files

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FVAdvection

Boundary conditions for pure advection

Example input syntax

Input Parameters

Required Parameters

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Input Files