LStableDirk3

Third order diagonally implicit Runge Kutta method (Dirk) with three stages.

This method can be expressed as a Runge-Kutta method with the following Butcher Tableau:

var element = document.getElementById("moose-equation-a215638d-58c3-4a52-a303-a8a7371371e6");katex.render("\\begin{array}{c|ccc} \\gamma & \\gamma \\\\ (1-\\gamma)/2 & \\gamma \\\\ 1 & (1/4)(-6\\gamma^2 + 16\\gamma - 1) & (1/4)(6\\gamma^2 - 20\\gamma + 5) & \\gamma \\\\ \\hline & (1/4)(-6\\gamma^2 + 16\\gamma - 1) & (1/4)(6\\gamma^2 - 20 \\gamma + 5) & \\gamma \\end{array}", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-90cc04db-b1f9-4181-ba8e-cc001b853e02");katex.render("\\gamma = -\\sqrt{2} \\cos(atan(\\sqrt{2}/4)/3)/2 + \\sqrt{6} \\sin(atan(\\sqrt{2}/4)/3)/2 + 1 \\approx 0.435866521508459", element, {displayMode:false,throwOnError:false});$

This method can be expressed as a Runge-Kutta method with the following Butcher Tableau:

var element = document.getElementById("moose-equation-c9154eb4-cca5-4402-ae12-79cc5772e501");katex.render("R(z) = \\dfrac{1.90128552647780115 z^2 + 2.46079651620301599 z - 8}{ 0.662446064957040178 z^3 - 4.55951098972521484 z^2 + 10.460796516203016 z - 8}", element, {displayMode:true,throwOnError:false});

The method is L-stable:

var element = document.getElementById("moose-equation-8794a9fa-492c-48ac-8b81-4ebefdd66ba5");katex.render("\\lim_{z->\\infty} R(z) = 0", element, {displayMode:true,throwOnError:false});

This method is derived in detail in Alexander (1977). Unlike BDF3, this method is L-stable and so may be more suitable for "stiff" problems.

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

References

R. Alexander. Diagonally implicit runge-kutta methods for stiff odes. SIAM J. Numer. Anal., 14(6):1006–1021, 1977.

@article{alexander1967,
    author = "Alexander, R.",
    title = "Diagonally implicit Runge-Kutta Methods for Stiff ODEs",
    journal = "SIAM J. Numer. Anal.",
    year = "1977",
    pages = "1006-1021",
    volume = "14",
    number = "6"
}

Input Parameters
References