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 $var element = document.getElementById("moose-equation-c9b71d6d-03f3-4120-924b-fb3a60b36061");katex.render("U", element, {displayMode:false,throwOnError:false});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-f64d352a-32af-4613-acb6-15e897175da3");katex.render("A", element, {displayMode:false,throwOnError:false});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-eef2210e-0026-4ebd-8abe-e961ebbe47c8");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false});

Using $var element = document.getElementById("moose-equation-a7d1cda6-d49f-4555-ac28-935097336270");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ for the current time step, and $var element = document.getElementById("moose-equation-f0b9981a-282d-4e93-942f-80bf0825fd16");katex.render("t", element, {displayMode:false,throwOnError:false});$ for the previous step, the implicit midpoint integration scheme can be written:

var element = document.getElementById("moose-equation-10a697f0-2360-439f-9567-64fea22976be");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});

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

var element = document.getElementById("moose-equation-0821a97b-b05f-4c53-aaa7-b8f791638b36");katex.render("\\begin{array}{c|c} 1/2 & 1/2 \\\\ \\hline & 1 \\end{array}", element, {displayMode:true,throwOnError:false});

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>
Unit:(no unit assumed)
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
Unit:(no unit assumed)
Controllable:No
Description:Set the enabled status of the MooseObject.

Advanced Parameters

Description
Input Parameters