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 $var element = document.getElementById("moose-equation-54fac521-c1b0-4da8-9e0e-3bdcee599ec9");katex.render("U", element, {displayMode:false,throwOnError:false});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-02f78c9f-6a04-440e-aa09-4ba22baf6794");katex.render("A", element, {displayMode:false,throwOnError:false});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-766b4cfe-9cd9-4f28-a330-ffa44b2cf156");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false});

Using $var element = document.getElementById("moose-equation-a9dc9d43-7832-40d4-a1f7-c8a690d80d32");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ for the current time step, and $var element = document.getElementById("moose-equation-13616083-a714-43fd-91fd-2c3fa6fcc953");katex.render("t", element, {displayMode:false,throwOnError:false});$ for the previous step, the implicit midpoint integration scheme can be written:

var element = document.getElementById("moose-equation-549e6911-655a-4a6c-a219-af38a3c78d42");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-30377816-d883-4b10-b558-be8fdaeb249e");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