lowEBMs.Packages.RK4¶
Runge-Kutta 4th order scheme¶
The lowEBMs.Packages.RK4
provides the numerical scheme to iteratively solve differential equations, hence the model equation which is parsed by lowEBMs.Packages.ModelEquation
, initialized with the configuration provided by lowEBMs.Packages.Configuration
.
For an example see How to use.
The increments \(k_1,..,k_4\) are obtained by solving the model equation, as defined in the physical background for the dynamical term \(\frac{dT}{dt}\). The increments differ in their choice of inital conditions (point of evaluation of the model equation). One iterative step always goes through a cycle of evaluating the model equation four times. It starts with the calculation of \(k_1\) at point \(y_0(t_0)\) with:
where \(f(t,y(t))\) is given by the deviation of \(y(t)\), hence \(\frac{dT}{dt} = \frac{1}{C}\cdot(R_{in}+R_{out}+...)\) at \(T_0 (t_0)\).
Now the scheme continues the following procedure:
As final step of one iterative step the weighted increment \(\phi\) is calculated by through:
to estimate \(y_1\) as final step of one iteration step: