FVAdvectedUpwind

Overview

This object provides first-order upwind interpolation for an advected quantity on a finite-volume face. The upwind state is selected using the sign of the face mass flux ( $var element = document.getElementById("moose-equation-8436154c-6d64-4aff-8636-d17a97bbd537");katex.render("m_f", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ ), yielding a monotone, Total Variation Diminishing (TVD) discretization that is robust near discontinuities at the cost of increased numerical diffusion (Moukalled et al. (2016), Harten (1997)).

Let $var element = document.getElementById("moose-equation-8e0b519c-c21f-4e9a-a71d-7917953dac7e");katex.render("\\phi_C", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ and $var element = document.getElementById("moose-equation-a648a895-f4ed-43e0-9f1d-353be49eb6b3");katex.render("\\phi_N", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ denote the cell-centered values on the element and neighbor sides of a face. Define the upwind value

var element = document.getElementById("moose-equation-790ce24f-98e6-49d6-892a-34381dfadfd7");katex.render("\\phi_U = \\begin{cases} \\phi_C & \\text{if } m_f \\ge 0,\\\\ \\phi_N & \\text{if } m_f < 0. \\end{cases}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Then the interpolated face value is simply

var element = document.getElementById("moose-equation-0205f9bc-2ad0-4812-964c-56d1bbe000de");katex.render("\\phi_f = \\phi_U.", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

In matrix form this corresponds to weights $var element = document.getElementById("moose-equation-0c9a060e-5b33-4f56-bf73-e6ec1cf2a546");katex.render("(w_C, w_N)\\in\\{(1,0),(0,1)\\}", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ chosen by the mass flux sign, so no gradients are required. This is a common baseline scheme for advection and is often used as the low-order component in higher-order schemes with limiting or deferred correction (see Limiters).

Example Syntax

Declare the interpolation method in [FVInterpolationMethods]:

[FVInterpolationMethods<<<{"href": "../../syntax/FVInterpolationMethods/index.html"}>>>]
  [upwind]
    type = FVAdvectedUpwind<<<{"description": "Upwind interpolation for advected quantities using the face mass flux sign.", "href": "FVAdvectedUpwind.html"}>>>
  []
[]

Use it in a linear FV advection kernel via "advected_interp_method_name":

[LinearFVKernels<<<{"href": "../../syntax/LinearFVKernels/index.html"}>>>]
  [advection]
    type = LinearFVAdvection<<<{"description": "Represents the matrix and right hand side contributions of an advection term in a partial differential equation.", "href": "../linearfvkernels/LinearFVAdvection.html"}>>>
    variable<<<{"description": "The name of the variable whose linear system this object contributes to"}>>> = u
    velocity<<<{"description": "Constant advection velocity"}>>> = "0.5 0 0"
    advected_interp_method_name<<<{"description": "Name of the FVInterpolationMethod to use for the advected quantity."}>>> = upwind
  []
[]

Input 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:No
Description:Set the enabled status of the MooseObject.

Input Files

(test/tests/postprocessors/linearfv_flux_integral/linearfv_flux_integral.i)
(test/tests/linearfvkernels/advection/advection-1d.i)
(test/tests/linearfvkernels/advection/diagonal-step-2d-delaunay.i)
(test/tests/linearfvkernels/advection/diagonal-step-2d.i)
(test/tests/linearfvkernels/advection/advection-2d.i)
(test/tests/linearfvkernels/advection/advection-2d-rz.i)

References

Ami Harten. High resolution schemes for hyperbolic conservation laws. Journal of computational physics, 135(2):260–278, 1997.

@article{harten1997,
    author = "Harten, Ami",
    title = "High resolution schemes for hyperbolic conservation laws",
    journal = "Journal of computational physics",
    volume = "135",
    number = "2",
    pages = "260--278",
    year = "1997",
    publisher = "Elsevier"
}

Fadl Moukalled, L Mangani, Marwan Darwish, and others. The finite volume method in computational fluid dynamics. Volume 6. Springer, 2016.

@book{moukalled2016finite,
    author = "Moukalled, Fadl and Mangani, L and Darwish, Marwan and others",
    title = "The finite volume method in computational fluid dynamics",
    volume = "6",
    year = "2016",
    publisher = "Springer"
}

(test/tests/linearfvkernels/advection/advection-1d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 2
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 1.0
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = upwind
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '0.5+0.5*x*x'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  multi_system_fixed_point_relaxation_factor = 1.0
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/linearfvkernels/advection/advection-1d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 2
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 1.0
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = upwind
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '0.5+0.5*x*x'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  multi_system_fixed_point_relaxation_factor = 1.0
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/postprocessors/linearfv_flux_integral/linearfv_flux_integral.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 100
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.25
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
[]

[LinearFVKernels]
  [diffusion]
    type = LinearFVDiffusion
    variable = u
    diffusion_coeff = 0.5
    use_nonorthogonal_correction = false
  []
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = '0.5 0 0'
    advected_interp_method_name = upwind
  []
[]

[LinearFVBCs]
  [left]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 1
  []
  [right]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = right
    functor = 0
  []
[]

[Postprocessors]
  [flux_left_total]
    type = BoundaryLinearFVFluxIntegral
    boundary = left
    linearfvkernels = 'diffusion advection'
  []
  [flux_left_diff]
    type = BoundaryLinearFVFluxIntegral
    boundary = left
    linearfvkernels = 'diffusion'
  []
  [flux_left_adv]
    type = BoundaryLinearFVFluxIntegral
    boundary = left
    linearfvkernels = 'advection'
  []
  [flux_right_total]
    type = BoundaryLinearFVFluxIntegral
    boundary = right
    linearfvkernels = 'diffusion advection'
  []
  [flux_right_diff]
    type = BoundaryLinearFVFluxIntegral
    boundary = right
    linearfvkernels = 'diffusion'
  []
  [flux_right_adv]
    type = BoundaryLinearFVFluxIntegral
    boundary = right
    linearfvkernels = 'advection'
  []
  [flux_left_sum_error]
    type = ParsedPostprocessor
    expression = 'f_total - f_diff - f_adv'
    pp_names = 'flux_left_total flux_left_diff flux_left_adv'
    pp_symbols = 'f_total f_diff f_adv'
  []
  [flux_right_sum_error]
    type = ParsedPostprocessor
    expression = 'f_total - f_diff - f_adv'
    pp_names = 'flux_right_total flux_right_diff flux_right_adv'
    pp_symbols = 'f_total f_diff f_adv'
  []
  [flux_balance]
    type = ParsedPostprocessor
    expression = 'f_left + f_right'
    pp_names = 'flux_left_total flux_right_total'
    pp_symbols = 'f_left f_right'
  []
  # These are the analytic solutions, when we refine we get closer and closer
  [analytic_flux_left_adv]
    type = ParsedPostprocessor
    expression = '-v'
    constant_names = 'v'
    constant_expressions = '0.5'
  []
  [analytic_flux_left_diff]
    type = ParsedPostprocessor
    expression = '-v / (exp(v / k * L) - 1)'
    constant_names = 'v k L'
    constant_expressions = '0.5 0.5 1'
  []
  [analytic_flux_right_adv]
    type = ParsedPostprocessor
    expression = '0'
  []
  [analytic_flux_right_diff]
    type = ParsedPostprocessor
    expression = 'v * exp(v / k * L) / (exp(v / k * L) - 1)'
    constant_names = 'v k L'
    constant_expressions = '0.5 0.5 1'
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-12
  l_abs_tol = 1e-12
  petsc_options_iname = '-pc_type'
  petsc_options_value = 'lu'
[]

[Outputs]
  csv = true
  execute_on = timestep_end
[]

(test/tests/linearfvkernels/advection/advection-1d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 2
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 1.0
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = upwind
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = '0.5*x'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = '0.5+0.5*x*x'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  multi_system_fixed_point_relaxation_factor = 1.0
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/linearfvkernels/advection/diagonal-step-2d-delaunay.i)

[Mesh]
  [file_mesh]
    type = FileMeshGenerator
    file = diagonal-step-2d-delaunay_in.e
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.5
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 0.75
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 0.75
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "1 1 0"
    advected_interp_method_name = muscl_venkat
  []
[]

[LinearFVBCs]
  [left_inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 0.5
  []
  [bottom_inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = bottom
    functor = 1
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = 'right top'
    use_two_term_expansion = false
  []
[]

[VectorPostprocessors]
  [diag_sample]
    type = LineValueSampler
    variable = u
    start_point = '1.0 0 0'
    end_point = '0 1.0 0'
    num_points = 101
    sort_by = id
    warn_discontinuous_face_values = false
    execute_on = TIMESTEP_END
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 200
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-12
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
  multi_system_fixed_point_relaxation_factor = 0.3
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

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

(test/tests/linearfvkernels/advection/diagonal-step-2d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 51
    ny = 51
  []
  # Prevent test diffing on distributed parallel element numbering
  allow_renumbering = false
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 0.5
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 0.75
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 0.75
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "1 1 0"
    advected_interp_method_name = nvd_minmod
  []
[]

[LinearFVBCs]
  [left_inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = left
    functor = 0.5
  []
  [bottom_inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = bottom
    functor = 1
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = 'right top'
    use_two_term_expansion = true
  []
[]

[VectorPostprocessors]
  [diag_sample]
    type = LineValueSampler
    variable = u
    start_point = '1.0 0 0'
    end_point = '0 1.0 0'
    num_points = 101
    sort_by = id
    warn_discontinuous_face_values = false
    execute_on = TIMESTEP_END
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 50
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-12
  multi_system_fixed_point = true
  multi_system_fixed_point_convergence = linear
  multi_system_fixed_point_relaxation_factor = 0.3
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

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

(test/tests/linearfvkernels/advection/advection-2d.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny= 1
    ymax = 0.5
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
  [muscl_venkat]
    type = FVAdvectedVenkatakrishnanDeferredCorrection
    deferred_correction_factor = 1.0
  []
  [nvd_vanleer]
    type = FVAdvectedVanLeerWeightBased
    blending_factor = 1.0
  []
  [nvd_minmod]
    type = FVAdvectedMinmodWeightBased
    blending_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.5 0 0"
    advected_interp_method_name = upwind
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left top bottom"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = '0.5*pi*sin(2*y*pi)*cos(x*pi)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = 'sin(x*pi)*sin(2*y*pi) + 1.5'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 1
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount -pc_factor_mat_solver_type -mat_mumps_icntl_14'
  petsc_options_value = 'lu       NONZERO               1e-12                     mumps                    50'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

(test/tests/linearfvkernels/advection/advection-2d-rz.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 2
    nx = 2
    ny= 1
    ymax = 0.5
  []
  coord_type = RZ
  rz_coord_axis = Y
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

[Variables]
  [u]
    type = MooseLinearVariableFVReal
    solver_sys = 'u_sys'
    initial_condition = 1.0
  []
[]

[FVInterpolationMethods]
  [upwind]
    type = FVAdvectedUpwind
  []
  [average]
    type = FVGeometricAverage
  []
[]

[LinearFVKernels]
  [advection]
    type = LinearFVAdvection
    variable = u
    velocity = "0.0 0.5 0"
    advected_interp_method_name = average
  []
  [source]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left right bottom"
    functor = analytic_solution
  []
  [outflow]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "top"
    use_two_term_expansion = true
  []
[]

[Functions]
  [source_func]
    type = ParsedFunction
    expression = '1.0*pi*sin(x*pi)*cos(2*y*pi)'
  []
  [analytic_solution]
    type = ParsedFunction
    expression = 'sin(x*pi)*sin(2*y*pi) + 1.5'
  []
[]

[Postprocessors]
  [error]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 10
    converge_at_max_iterations = true
  []
[]

[Executioner]
  type = Steady
  system_names = u_sys
  l_tol = 1e-10
  multi_system_fixed_point=true
  multi_system_fixed_point_convergence=linear
  petsc_options_iname = '-pc_type -pc_factor_shift_type -pc_factor_shift_amount'
  petsc_options_value = 'lu       NONZERO               1e-10'
[]

[Outputs]
  [csv]
    type = CSV
    execute_on = FINAL
  []
[]

Overview
Example Syntax
Input Parameters
Input Files
References