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 $var element = document.getElementById("moose-equation-6fb19d79-2a0f-45fa-a203-cc7a6af82873");katex.render("U", element, {displayMode:false,throwOnError:false});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-e0c587ee-c863-4807-bb79-aee8ba872023");katex.render("A", element, {displayMode:false,throwOnError:false});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-c0ae99ee-bae7-4533-81c3-5047d1627325");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false});

Using $var element = document.getElementById("moose-equation-4398d195-82ac-4ad0-9d24-8155c3b8078c");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ for the current time step, and $var element = document.getElementById("moose-equation-98066228-0650-48a8-92c7-a13b8ca909ef");katex.render("t", element, {displayMode:false,throwOnError:false});$ for the previous step, the explicit midpoint integration scheme can be written:

var element = document.getElementById("moose-equation-62a5b96e-b8f2-436a-9737-6971db70d72d");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-b303abe3-6b9a-455d-9bd9-5bf517b89b68");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