CrankNicolson

Crank-Nicolson time integrator.

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

Description

With $var element = document.getElementById("moose-equation-0c45968c-59ea-4211-a08b-967af180fcd4");katex.render("U", element, {displayMode:false,throwOnError:false});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-57bde162-bd4d-406b-bafd-d06c53702857");katex.render("A", element, {displayMode:false,throwOnError:false});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-e3b40fbc-7b4c-4bb1-affd-39e380bed5b9");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false});

Using $var element = document.getElementById("moose-equation-41d574c8-3ab1-4a83-a191-507f1186caed");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ for the current time step, and $var element = document.getElementById("moose-equation-a8fe8010-e71c-48e0-9def-57ad20987ed2");katex.render("t", element, {displayMode:false,throwOnError:false});$ for the previous step, the Crank Nicolson time integration scheme can be written:

var element = document.getElementById("moose-equation-9f543385-7298-4fef-9a4e-20377b3e404e");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-472e159e-0e34-4f06-a173-6e07685c35e3");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