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:

γγ(1γ)/2γ1(1/4)(6γ2+16γ1)(1/4)(6γ220γ+5)γ(1/4)(6γ2+16γ1)(1/4)(6γ220γ+5)γ\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}

where γ=2cos(atan(2/4)/3)/2+6sin(atan(2/4)/3)/2+10.435866521508459\gamma = -\sqrt{2} \cos(atan(\sqrt{2}/4)/3)/2 + \sqrt{6} \sin(atan(\sqrt{2}/4)/3)/2 + 1 \approx 0.435866521508459

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

R(z)=1.90128552647780115z2+2.46079651620301599z80.662446064957040178z34.55951098972521484z2+10.460796516203016z8R(z) = \dfrac{1.90128552647780115 z^2 + 2.46079651620301599 z - 8}{ 0.662446064957040178 z^3 - 4.55951098972521484 z^2 + 10.460796516203016 z - 8}

The method is L-stable:

limz>R(z)=0\lim_{z->\infty} R(z) = 0

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

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