CrankNicolson

Crank-Nicolson time integrator.

The Crank Nicolson time integration scheme is a second-order implicit method. It is unconditionally stable.

Description

With $var element = document.getElementById("moose-equation-9418ac14-9bf0-4082-ab27-d0a1fcb7dc43");katex.render("U", element, {displayMode:false,throwOnError:false});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-4456c407-dc2b-4065-9ab9-3b51e8fc7a74");katex.render("A", element, {displayMode:false,throwOnError:false});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-a9ed142b-1c33-4590-881a-55492466783c");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false});

Using $var element = document.getElementById("moose-equation-4c917569-5c2a-4431-be9f-6db1299725eb");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ for the current time step, and $var element = document.getElementById("moose-equation-8a61b71f-1552-4ba2-a1d1-6423283c1993");katex.render("t", element, {displayMode:false,throwOnError:false});$ for the previous step, the Crank Nicolson time integration scheme can be written:

var element = document.getElementById("moose-equation-71abf513-f192-4afd-84b1-0597be86263b");katex.render("U(t+\\Delta t) = U(t) + \\Delta t \\dfrac{1}{2} \\left( A(t, U(t)) + A(t+\\Delta t, U(t+\\Delta t)) \\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-5a9ccb1c-ed21-4b20-bdbf-4ee6eaec1f97");katex.render("\\begin{array}{c|cc} 0 & 0 & 0 \\\\ 1 & 1/2 & 1/2 \\\\ \\hline & 1/2 & 1/2 \\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