AStableDirk4

Fourth-order diagonally implicit Runge Kutta method (Dirk) with three stages plus an update.

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

var element = document.getElementById("moose-equation-0ff5701a-e6c5-4df1-bd20-6b1ed3dc3ed9");katex.render("\\begin{array}{c|ccc} \\gamma & \\gamma & 0 & 0 \\\\ 1/2 & 1/2 - \\gamma & \\gamma & 0 \\\\ 1 - \\gamma & 2 \\gamma & 1 - 4 \\gamma & \\gamma \\\\ \\hline & \\dfrac{1}{24(1/2-\\gamma)} & 1 - \\dfrac{1}{12(1/2-\\gamma)^2} & \\dfrac{1}{24(1/2-\\gamma)^2} \\end{array}", element, {displayMode:true,throwOnError:false});

where $var element = document.getElementById("moose-equation-49dd5e57-7b9e-44a7-b9d7-281c27788fb5");katex.render("\\gamma = 1/2 + \\dfrac{\\sqrt{3}}{3} \\cos( \\dfrac{\\pi}{18}) \\approx 1.06857902130162881", element, {displayMode:false,throwOnError:false});$

The stability function for this method is:

var element = document.getElementById("moose-equation-49c3fc40-f035-4fec-8cf0-e397a6b184c1");katex.render("R(z) = \\dfrac{-0.76921266689461 z^3 - 0.719846311954034 z^2 + 2.20573706179581 z - 1.0}{ 1.22016884572716 z^3 - 3.42558337748497 z^2 + 3.20573706551682 z - 1.0}", element, {displayMode:true,throwOnError:false});

The method is not L-stable; it is only A-stable:

var element = document.getElementById("moose-equation-d220105c-8ead-4b88-a541-2eb3beb9a52b");katex.render("\\lim_{z->\\infty} R(z) = -0.630414937726258", element, {displayMode:true,throwOnError:false});

Notes

Method is originally due to Crouzeix (1975)
Since $var element = document.getElementById("moose-equation-6e4dc7dd-ca2e-41f9-8783-2a431211eb59");katex.render("\\gamma", element, {displayMode:false,throwOnError:false});$ is slightly larger than $var element = document.getElementById("moose-equation-e5e32939-1958-4d65-a28d-d0454813a8a0");katex.render("1", element, {displayMode:false,throwOnError:false});$ , the first stage involves evaluation of the non-time residuals for $var element = document.getElementById("moose-equation-89632559-ab8f-465f-8d7b-69a41acc2ffa");katex.render("t > t_n+\\Delta t", element, {displayMode:false,throwOnError:false});$ , while the third stage involves evaluation of the non-time residual for $var element = document.getElementById("moose-equation-edb191f8-27cc-495b-a849-e1dcc29d237f");katex.render("t < t_n", element, {displayMode:false,throwOnError:false});$ , which may present an issue for the first timestep (if e.g. material properties or forcing functions are not defined for $var element = document.getElementById("moose-equation-86987283-6869-4d3e-ba87-2b4590719fe5");katex.render("t<0", element, {displayMode:false,throwOnError:false});$ . We could handle this by using an alternate (more expensive) method in the first timestep, or by using a lower-order method for the first timestep and then switching to this method for subsequent timesteps.

Input Parameters

safe_startTrueIf true, use LStableDirk4 to bootstrap this method.
Default:True
C++ Type:bool
Unit:(no unit assumed)
Controllable:No
Description:If true, use LStableDirk4 to bootstrap this method.
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

M. Crouzeix. Sur l'approximation des equations differentielles operationelles lineaires par des methodes de Runge Kutta. PhD thesis, Universite Paris VI, Paris, 1975.

@phdthesis{crouzeix1975,
    author = "Crouzeix, M.",
    title = "Sur l'approximation des equations differentielles operationelles lineaires par des methodes de Runge Kutta",
    school = "Universite Paris VI, Paris",
    year = "1975"
}

Notes
Input Parameters
References