FVGreenGaussGradient

Overview

This method computes gradients for cell-centered linear finite-volume variables using the Green-Gauss theorem Moukalled et al. (2016). For a cell $var element = document.getElementById("moose-equation-48c4f7fc-e070-43f9-8527-148c06e9e741");katex.render("C", element, {displayMode:false,throwOnError:false});$ with volume $var element = document.getElementById("moose-equation-0239f756-71e0-4f3f-8d62-b9cc68422bd2");katex.render("V_C", element, {displayMode:false,throwOnError:false});$ , the method approximates the gradient of a cell-centered field $var element = document.getElementById("moose-equation-14d0920b-6ad1-4ed3-8895-e11d72014779");katex.render("\\phi", element, {displayMode:false,throwOnError:false});$ as

var element = document.getElementById("moose-equation-bb8443c9-67fd-4e91-8a2a-539a305b054e");katex.render("\\nabla \\phi_C \\approx \\frac{1}{V_C}\\sum_f \\phi_f \\mathbf{S}_f,", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-f05213a0-06aa-4ab0-ada1-3c8b1303ea54");katex.render("f", element, {displayMode:false,throwOnError:false});$ ranges over the cell faces, $var element = document.getElementById("moose-equation-58b36d0f-df8f-4b94-a1e6-1d835562c81f");katex.render("\\phi_f", element, {displayMode:false,throwOnError:false});$ is the face value interpolated from adjacent cell-centered values and boundary data, and $var element = document.getElementById("moose-equation-bedb37df-9bcc-4feb-a5ac-7638fc214b45");katex.render("\\mathbf{S}_f", element, {displayMode:false,throwOnError:false});$ is the outward face area vector.

Use this method when a linear FV variable or a gradient-based interpolation method needs gradients. MOOSE also provides two built-in method names for convenient input into gradient method parameters:

green-gauss, equivalent to an FVGreenGaussGradient with limiter = none
green-gauss-venkatakrishnan, equivalent to an FVGreenGaussGradient with limiter = venkatakrishnan

Named methods in [FVGradientMethods] are useful when several variables or interpolation methods should use the same gradient settings.

Limiters

The optional "limiter" parameter controls whether MOOSE limits the Green-Gauss gradient before using it. With limiter = none, the Green-Gauss result is used directly. With limiter = venkatakrishnan, MOOSE first computes the Green-Gauss gradient and then applies the Venkatakrishnan limiter Venkatakrishnan (1993). A limiter can reduce overshoots near steep solution changes.

Example Syntax

Declare a named Green-Gauss gradient method in [FVGradientMethods]:

[FVGradientMethods<<<{"href": "../../syntax/FVGradientMethods/index.html"}>>>]
  [gg]
    type = FVGreenGaussGradient<<<{"description": "Green-Gauss cell-centered gradient method.", "href": "FVGreenGaussGradient.html"}>>>
  []
[]

Use the method as the default gradient method for a linear FV variable through "gradient_method":

[Variables<<<{"href": "../../syntax/Variables/index.html"}>>>]
  [u]
    type = MooseLinearVariableFVReal<<<{"description": "Base class for Moose variables. This should never be the terminal object type", "href": "../variables/MooseLinearVariableFV.html"}>>>
    solver_sys<<<{"description": "If this variable is a solver variable, this is the solver system to which it should be added."}>>> = 'u_sys'
    initial_condition<<<{"description": "Specifies a constant initial condition for this variable"}>>> = 1.0
    gradient_method<<<{"description": "Default gradient computation method to register when a consumer requests gradients from this variable. This may be a built-in method name like 'green-gauss' or 'green-gauss-venkatakrishnan', or the name of an object in [FVGradientMethods]."}>>> = gg
  []
[]

The same named method can also be used by a gradient-based interpolation method, such as FVAdvectedMUSCLDeferredCorrection:

[FVInterpolationMethods<<<{"href": "../../syntax/FVInterpolationMethods/index.html"}>>>]
  [muscl]
    type = FVAdvectedMUSCLDeferredCorrection<<<{"description": "MUSCL reconstruction with cell gradients from a named gradient method using deferred correction.", "href": "../fvinterpolationmethods/FVAdvectedMUSCLDeferredCorrection.html"}>>>
    gradient_method<<<{"description": "Gradient method used to compute cell gradients for the high-order reconstruction."}>>> = gg
    deferred_correction_factor<<<{"description": "Scales the deferred correction strength; 0 gives pure upwind (no deferred correction), 1 gives full deferred correction. Values < 1 can improve fixed point robustness."}>>> = 1.0
  []
[]

Input Parameters

limiternoneLimiter to apply to gradients produced by this method.
Default:none
C++ Type:MooseEnum
Options:none, venkatakrishnan
Controllable:No
Description:Limiter to apply to gradients produced by this method.

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

Advanced Parameters

References

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

BibTeX

@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"
}

RIS

TY  - BOOK
AU  - Moukalled, Fadl
AU  - Mangani, L
AU  - Darwish, Marwan
AU  - others
TI  - The finite volume method in computational fluid dynamics
PY  - 2016
VL  - 6
PB  - Springer
ER  -

Plain Text

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

Venkat Venkatakrishnan. On the accuracy of limiters and convergence to steady state solutions. In 31st Aerospace Sciences Meeting, 880. 1993.

BibTeX

@inproceedings{venkatakrishnan1993,
    author = "Venkatakrishnan, Venkat",
    title = "On the accuracy of limiters and convergence to steady state solutions",
    booktitle = "31st Aerospace Sciences Meeting",
    pages = "880",
    year = "1993"
}

RIS

TY  - CPAPER
AU  - Venkatakrishnan, Venkat
TI  - On the accuracy of limiters and convergence to steady state solutions
T2  - 31st Aerospace Sciences Meeting
PY  - 1993
SP  - 880
ER  -

Plain Text

Venkat Venkatakrishnan. On the accuracy of limiters and convergence to steady state solutions. In 31st Aerospace Sciences Meeting, 880. 1993.

(../moose/test/tests/variables/linearfv/shared-gradient-method.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 4
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

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

[FVGradientMethods]
  [gg]
    type = FVGreenGaussGradient
  []
[]

[FVInterpolationMethods]
  [muscl]
    type = FVAdvectedMUSCLDeferredCorrection
    gradient_method = gg
    deferred_correction_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection_u]
    type = LinearFVAdvection
    variable = u
    velocity = "1 0 0"
    advected_interp_method_name = muscl
  []
  [source_u]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
  [advection_v]
    type = LinearFVAdvection
    variable = v
    velocity = "1 0 0"
    advected_interp_method_name = muscl
  []
  [source_v]
    type = LinearFVSource
    variable = v
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow_u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = analytic_solution
  []
  [outflow_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
  [inflow_v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = v
    boundary = "left"
    functor = analytic_solution
  []
  [outflow_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = v
    boundary = "right"
    use_two_term_expansion = false
  []
[]

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

[Postprocessors]
  [error_u]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [error_v]
    type = ElementL2FunctorError
    approximate = v
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 8
    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
  []
[]

(../moose/test/tests/variables/linearfv/shared-gradient-method.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 4
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

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

[FVGradientMethods]
  [gg]
    type = FVGreenGaussGradient
  []
[]

[FVInterpolationMethods]
  [muscl]
    type = FVAdvectedMUSCLDeferredCorrection
    gradient_method = gg
    deferred_correction_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection_u]
    type = LinearFVAdvection
    variable = u
    velocity = "1 0 0"
    advected_interp_method_name = muscl
  []
  [source_u]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
  [advection_v]
    type = LinearFVAdvection
    variable = v
    velocity = "1 0 0"
    advected_interp_method_name = muscl
  []
  [source_v]
    type = LinearFVSource
    variable = v
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow_u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = analytic_solution
  []
  [outflow_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
  [inflow_v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = v
    boundary = "left"
    functor = analytic_solution
  []
  [outflow_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = v
    boundary = "right"
    use_two_term_expansion = false
  []
[]

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

[Postprocessors]
  [error_u]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [error_v]
    type = ElementL2FunctorError
    approximate = v
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 8
    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
  []
[]

(../moose/test/tests/variables/linearfv/shared-gradient-method.i)

[Mesh]
  [gmg]
    type = GeneratedMeshGenerator
    dim = 1
    nx = 4
  []
[]

[Problem]
  linear_sys_names = 'u_sys'
[]

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

[FVGradientMethods]
  [gg]
    type = FVGreenGaussGradient
  []
[]

[FVInterpolationMethods]
  [muscl]
    type = FVAdvectedMUSCLDeferredCorrection
    gradient_method = gg
    deferred_correction_factor = 1.0
  []
[]

[LinearFVKernels]
  [advection_u]
    type = LinearFVAdvection
    variable = u
    velocity = "1 0 0"
    advected_interp_method_name = muscl
  []
  [source_u]
    type = LinearFVSource
    variable = u
    source_density = source_func
  []
  [advection_v]
    type = LinearFVAdvection
    variable = v
    velocity = "1 0 0"
    advected_interp_method_name = muscl
  []
  [source_v]
    type = LinearFVSource
    variable = v
    source_density = source_func
  []
[]

[LinearFVBCs]
  [inflow_u]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = u
    boundary = "left"
    functor = analytic_solution
  []
  [outflow_u]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = u
    boundary = "right"
    use_two_term_expansion = false
  []
  [inflow_v]
    type = LinearFVAdvectionDiffusionFunctorDirichletBC
    variable = v
    boundary = "left"
    functor = analytic_solution
  []
  [outflow_v]
    type = LinearFVAdvectionDiffusionOutflowBC
    variable = v
    boundary = "right"
    use_two_term_expansion = false
  []
[]

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

[Postprocessors]
  [error_u]
    type = ElementL2FunctorError
    approximate = u
    exact = analytic_solution
    execute_on = FINAL
  []
  [error_v]
    type = ElementL2FunctorError
    approximate = v
    exact = analytic_solution
    execute_on = FINAL
  []
  [h]
    type = AverageElementSize
    execute_on = FINAL
  []
[]

[Convergence]
  [linear]
    type = IterationCountConvergence
    max_iterations = 8
    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
  []
[]

Overview
Limiters
Example Syntax
Input Parameters
References