ExplicitTVDRK2

Explicit TVD (total-variation-diminishing) second-order Runge-Kutta time integration method.

Description

The method Gottlieb and Shu (1998) consists of two stages:

Stage 1.

var element = document.getElementById("moose-equation-8c9bf097-87aa-40c3-87ef-a382cc52d7f7");katex.render("R_{NL} = M(U^{(1)}-U^n)/\\Delta t - F(t^n,U^n)", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

var element = document.getElementById("moose-equation-d6ffbf80-1fe7-4186-8e2b-e7a03252738d");katex.render("t^{(1)} = t^{n} + \\Delta t = t^{n+1}", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

Stage 2.

var element = document.getElementById("moose-equation-b75e0446-70af-4f48-9edc-2366f75b2e48");katex.render("R_{NL} = M(2U^{(2)}-U^{(1)}-U^n)/(2\\Delta t) - (1/2)F(t^{(1)},U^{(1)})", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});

var element = document.getElementById("moose-equation-0d7aa9fc-9900-46fa-8704-d6532f1b77b0");katex.render("U^{n+1} = U^{(2)}", 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 method requires two mass matrix (linear) system solves per timestep. Although strictly speaking these are "two stage" methods, we treat the "update" step as a third stage, since in finite element analysis the update step requires a mass matrix solve.

warning

To use the explicit TimeIntegrators derived from this method, you must generally add "implicit=false" to the Kernels, Materials, etc. used in your simulation, so that MOOSE evaluates them correctly! An important exception are TimeDerivative kernels, which should never be marked "implicit=false".

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/time_integrators/tvdrk2/1d-linear.i)
(test/tests/time_integrators/tvdrk2/2d-quadratic.i)

References

S. Gottlieb and C. W. Shu. Total variation diminishing runge-kutta schemes. Mathematics of computation of the American Mathematical Society, 67(221):73–85, 1998.

@article{gottlieb1998,
    author = "Gottlieb, S. and Shu, C. W.",
    title = "Total variation diminishing Runge-Kutta schemes",
    year = "1998",
    journal = "Mathematics of computation of the American Mathematical Society",
    volume = "67",
    number = "221",
    pages = "73-85"
}

(test/tests/time_integrators/tvdrk2/1d-linear.i)

[Mesh]
  type = GeneratedMesh
  dim = 1
  xmin = -1
  xmax = 1
  nx = 20
  elem_type = EDGE2
[]

[Functions]
  [./ic]
    type = ParsedFunction
    expression = 0
  [../]

  [./forcing_fn]
    type = ParsedFunction
    expression = x
  [../]

  [./exact_fn]
    type = ParsedFunction
    expression = t*x
  [../]
[]

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

[Kernels]
  [./ie]
    type = TimeDerivative
    variable = u
    implicit = true
  [../]

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

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

[ICs]
  [./u_ic]
    type = FunctionIC
    variable = u
    function = ic
  [../]
[]

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

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

[Executioner]
  type = Transient

  [./TimeIntegrator]
    type = ExplicitTVDRK2
  [../]
  solve_type = 'LINEAR'

  start_time = 0.0
  num_steps = 10
  dt = 0.001
  l_tol = 1e-15
[]

[Outputs]
  exodus = true
  perf_graph = true
[]

(test/tests/time_integrators/tvdrk2/2d-quadratic.i)

[Mesh]
  type = GeneratedMesh
  dim = 2
  xmin = -1
  xmax = 1
  ymin = -1
  ymax = 1
  nx = 20
  ny = 20
  elem_type = QUAD9
[]

[Functions]
  [./ic]
    type = ParsedFunction
    expression = 0
  [../]

  [./forcing_fn]
    type = ParsedFunction
    expression = 2*t*((x*x)+(y*y))-(4*t*t)
  [../]

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

[Variables]
  [./u]
    order = SECOND
    family = LAGRANGE
  [../]
[]

[Kernels]
  [./ie]
    type = TimeDerivative
    variable = u
    implicit = true
  [../]

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

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

[ICs]
  [./u_ic]
    type = FunctionIC
    variable = u
    function = ic
  [../]
[]

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

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

[Executioner]
  type = Transient

  [./TimeIntegrator]
    type = ExplicitTVDRK2
  [../]
  solve_type = 'LINEAR'

  start_time = 0.0
  num_steps = 10
  dt = 0.0001
  l_tol = 1e-8
[]

[Outputs]
  exodus = true
  perf_graph = true
[]

Description
Input Parameters
Input Files
References