ImplicitMidpoint

Second-order Runge-Kutta (implicit midpoint) time integration.

The implicit midpoint method is second-order accurate. As a Gauss-Legendre method it is A-stable.

Description

With $var element = document.getElementById("moose-equation-a2f58d5d-1161-42b3-83b6-cde69e57289c");katex.render("U", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-69bc67e6-c574-4214-8ddb-7a863cc99cf1");katex.render("A", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-62670e39-8191-4d5e-bce8-2b99b91bc3d5");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Using $var element = document.getElementById("moose-equation-a1f6c349-9179-46cf-8e9d-c61ed4a75fb9");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ for the current time step, and $var element = document.getElementById("moose-equation-4e179337-6aa0-421a-80ef-ddaf87af63a6");katex.render("t", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ for the previous step, the implicit midpoint integration scheme can be written:

var element = document.getElementById("moose-equation-97d62396-baa1-4172-92ae-0f24efb210e7");katex.render("U(t+\\Delta t) = U(t) + \\Delta t A \\left(t+\\Delta t/2, \\dfrac{\\Delta t}{2} \\left( U(t) + U(t+\\Delta t) \\right) \\right)", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

This method can be expressed as a Runge-Kutta method with the following Butcher Tableau:

var element = document.getElementById("moose-equation-28471e0c-99c0-4322-8856-088f2c89d558");katex.render("\\begin{array}{c|c} 1/2 & 1/2 \\\\ \\hline & 1 \\end{array}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Input Parameters

variablesA subset of the variables that this time integrator should be applied to
C++ Type:std::vector<VariableName>
Unit:(no unit assumed)
Controllable:No
Description:A subset of the variables that this time integrator should be applied to

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

Input Files

(test/tests/time_integrators/convergence/implicit_convergence.i)
(test/tests/time_integrators/scalar/scalar.i)

(test/tests/time_integrators/convergence/implicit_convergence.i)

[Mesh]
  type = GeneratedMesh
  dim  = 2
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  nx   = 4
  ny   = 4
  elem_type = QUAD9
[]

[Variables]
  active = 'u'

  [./u]
    order = SECOND
    family = LAGRANGE

    [./InitialCondition]
      type = ConstantIC
      value = 0
    [../]
  [../]
[]

[Functions]
  [./exact_fn]
    type = ParsedFunction
    expression = t*t*t*((x*x)+(y*y))
  [../]

  [./forcing_fn]
    type = ParsedFunction
    expression = 3*t*t*((x*x)+(y*y))-(4*t*t*t)
  [../]
[]

[Kernels]
  active = 'diff ie ffn'

  [./ie]
    type = TimeDerivative
    variable = u
  [../]

  [./diff]
    type = Diffusion
    variable = u
  [../]

  [./ffn]
    type = BodyForce
    variable = u
    function = forcing_fn
  [../]
[]

[BCs]
  [./all]
    type = FunctionDirichletBC
    variable = u
    boundary = '0 1 2 3'
    function = exact_fn
    preset = false
  [../]
[]

[Postprocessors]
  [./l2_err]
    type = ElementL2Error
    variable = u
    function = exact_fn
  [../]
[]

[Executioner]
  type = Transient

  solve_type = 'PJFNK'
  petsc_options_iname = '-pc_type -pc_hypre_type'
  petsc_options_value = 'hypre boomeramg'

  start_time = 0.0
  end_time = 1.0
  dt = 0.0625

 [./TimeIntegrator]
   type = ImplicitMidpoint
 [../]
[]

[Outputs]
  execute_on = 'initial timestep_end'
  exodus = true
[]

(test/tests/time_integrators/scalar/scalar.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = 0
  xmax = 1
  ymin = 0
  ymax = 1
  nx = 1
  ny = 1
  elem_type = QUAD4
[]

[Variables]
  [./n]
    family = SCALAR
    order = FIRST
    initial_condition = 1
  [../]
[]

[ScalarKernels]
  [./dn]
    type = ODETimeDerivative
    variable = n
  [../]
  [./ode1]
    type = ParsedODEKernel
    expression = '-n'
    variable = n
    # implicit = false
  [../]
[]

[Executioner]
  type = Transient
  [./TimeIntegrator]
    # type = ImplicitEuler
    # type = BDF2
    type = CrankNicolson
    # type = ImplicitMidpoint
    # type = LStableDirk2
    # type = LStableDirk3
    # type = LStableDirk4
    # type = AStableDirk4
    #
    # Explicit methods
    # type = ExplicitEuler
    # type = ExplicitMidpoint
    # type = Heun
    # type = Ralston
  [../]
  start_time = 0
  end_time = 1
  dt = 0.001
  dtmin = 0.001 # Don't allow timestep cutting
  solve_type = 'PJFNK'
  nl_max_its = 2
  nl_abs_tol = 1.e-12 # This is an ODE, so nl_abs_tol makes sense.
[]

[Functions]
  [./exact_solution]
    type = ParsedFunction
    expression = exp(t)
  [../]
[]

[Postprocessors]
  [./error_n]
    # Post processor that computes the difference between the computed
    # and exact solutions.  For the exact solution used here, the
    # error at the final time should converge at O(dt^p), where p is
    # the order of the method.
    type = ScalarL2Error
    variable = n
    function = exact_solution
    # final is not currently supported for Postprocessor execute_on...
    # execute_on = 'final'
  [../]
[]

[Outputs]
  csv = true
[]

Description
Input Parameters
Input Files