ExplicitMidpoint

Time integration using the explicit midpoint method.

The explicit midpoint method is second-order accurate in time. It is a two-step method and a special case of the 2nd-order Runge-Kutta method.

Description

With $var element = document.getElementById("moose-equation-39138392-65b6-46e3-85a4-f6c5676a78b0");katex.render("U", element, {displayMode:false,throwOnError:false});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-0f95f940-877a-477f-9f01-0364541d7cf4");katex.render("A", element, {displayMode:false,throwOnError:false});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-f6912496-7e14-489e-8d98-36b33550f388");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false});

Using $var element = document.getElementById("moose-equation-82f02599-6ad6-424c-9c41-a50e69801c43");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ for the current time step, and $var element = document.getElementById("moose-equation-a9ccc1e8-dd9e-4502-b4ad-ca96011f7b94");katex.render("t", element, {displayMode:false,throwOnError:false});$ for the previous step, the explicit midpoint integration scheme can be written:

var element = document.getElementById("moose-equation-c6a93f3c-7509-4e9e-a9f7-08e4cab06b4f");katex.render("U(t+\\Delta t) = U(t) + \\Delta t A(t+\\Delta t/2, U(t) + \\dfrac{\\Delta t}{2} A(t,U(t)))", 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-3fe43bc1-697f-4aeb-8dc0-6526e2d9a477");katex.render("\\begin{array}{c|cc} 0 & 0 \\\\ 1/2 & 1/2 & 0 \\\\ \\hline & 0 & 1 \\end{array}", element, {displayMode:true,throwOnError:false});

warning

All kernels except time-(derivative)-kernels should have the parameter implicit=false to use this time integrator.

warning

ExplicitRK2-derived TimeIntegrators (ExplicitMidpoint, Heun, Ralston) and other multistage TimeIntegrators are known not to work with Materials/AuxKernels that accumulate 'state' and should be used with caution.

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