Ralston

Ralston's time integration method.

Ralston's time integration method is second-order accurate in time. It is a two-step explicit method and a special case of the 2nd-order Runge-Kutta method. It is obtained through an error minimization process and has been shown to outperform other 2nd-order explicit Runge-Kutta methods, see Ralston (1962).

Description

With $var element = document.getElementById("moose-equation-2b8df22a-64a7-4908-a3a8-582b77959506");katex.render("U", element, {displayMode:false,throwOnError:false});$ , the vector of nonlinear variables, and $var element = document.getElementById("moose-equation-b8f1d563-8cc3-47be-af25-dbcc3ffa7c00");katex.render("A", element, {displayMode:false,throwOnError:false});$ , a nonlinear operator, we write the PDE of interest as:

var element = document.getElementById("moose-equation-88c7f190-ad6e-4614-a286-4b87943e2520");katex.render("\\dfrac{\\partial U}{\\partial t} = A(t, U(t))", element, {displayMode:true,throwOnError:false});

Using $var element = document.getElementById("moose-equation-e0d6041b-6b2e-4625-bebf-5ea714d1ff79");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ for the current time step and $var element = document.getElementById("moose-equation-ad134d18-2e99-48c5-a409-e69384f01545");katex.render("t", element, {displayMode:false,throwOnError:false});$ for the previous step, Ralston's method can be written:

var element = document.getElementById("moose-equation-9416206d-d2ec-427b-b419-670ceb39cdb9");katex.render("U(t+\\Delta t) = U(t) + \\dfrac{\\Delta t}{4} \\left(A(t, U(t)\\right) + \\dfrac{3\\Delta t}{4} A \\left(t + \\dfrac{2\\Delta t}{3},U(t) + \\dfrac{2\\Delta t}{3} A(t, U(t)) \\right)", element, {displayMode:true,throwOnError:false});

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

var element = document.getElementById("moose-equation-da5e1fb7-6506-4ba9-bab5-8ef75c20bd83");katex.render("\\begin{array}{c|cc} 0 & 0 \\\\ 2/3 & 2/3 & 0 \\\\ \\hline & 1/4 & 3/4 \\end{array}", element, {displayMode:true,throwOnError:false});

warning

All kernels except time-(derivative)-kernels should have the parameter implicit=false to use this time integrator.

warning

ExplicitRK2-derived TimeIntegrators ExplicitMidpoint, Heun, Ralston) and other multistage TimeIntegrators are known not to work with Materials/AuxKernels that accumulate 'state' and should be used with caution.

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

Anthony Ralston. Runge-kutta methods with minimum error bounds. Math. Comput., 80:431–437, 1962. doi:10.1090/S0025-5718-1962-0150954-0.

@article{ralston1962,
    author = "Ralston, Anthony",
    title = "Runge-Kutta Methods with Minimum Error Bounds",
    year = "1962",
    journal = "Math. Comput.",
    volume = "80",
    pages = "431–437",
    doi = "10.1090/S0025-5718-1962-0150954-0"
}

Description
Input Parameters
References