LStableDirk2

Second order diagonally implicit Runge Kutta method (Dirk) with two stages.

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

var element = document.getElementById("moose-equation-cc1de091-2e45-450e-859b-cbc59a787969");katex.render("\\begin{array}{c|cc} \\alpha & \\alpha\\\\ 1 & 1 - \\alpha & \\alpha \\\\ \\hline & 1 - \\alpha & \\alpha \\end{array}", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-918fa2e5-64e9-49aa-9503-b03711468b11");katex.render("\\alpha = 1 - \\sqrt{2}/2 \\approx 0.29289", element, {displayMode:false,throwOnError:false});$

The stability function for this method is:

var element = document.getElementById("moose-equation-de632bb4-aaf9-4fa1-b344-df1cc97c722f");katex.render("R(z) = 4 \\dfrac{-z(-\\sqrt{2} + 2) + z + 1}{z^2(-\\sqrt{2} + 2)^2 - 4z(-\\sqrt{2} + 2) + 4}", element, {displayMode:true,throwOnError:false});

The method is L-stable:

var element = document.getElementById("moose-equation-7371fec1-c17f-455d-aa15-b35609f5b861");katex.render("\\lim_{z->\\infty} R(z) = 0", element, {displayMode:true,throwOnError:false});

Notes

This method is derived in detail in Alexander (1977). This method is more expensive than Crank-Nicolson, but has the advantage of being L-stable (the same type of stability as the implicit Euler method) 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"
}

Notes
Input Parameters
References