ImplicitMidpoint

Second-order Runge-Kutta (implicit midpoint) time integration.

The implicit midpoint method is second-order accurate. As a Gauss-Legendre method it is A-stable.

Description

With UU, the vector of nonlinear variables, and AA, a nonlinear operator, we write the PDE of interest as:

Ut=A(t,U(t))\dfrac{\partial U}{\partial t} = A(t, U(t))

Using t+Δtt+\Delta t for the current time step, and tt for the previous step, the implicit midpoint integration scheme can be written:

U(t+Δt)=U(t)+ΔtA(t+Δt/2,Δt2(U(t)+U(t+Δt)))U(t+\Delta t) = U(t) + \Delta t A \left(t+\Delta t/2, \dfrac{\Delta t}{2} \left( U(t) + U(t+\Delta t) \right) \right)

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

1/21/21\begin{array}{c|c} 1/2 & 1/2 \\ \hline & 1 \end{array}

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