Runge-Kutta-Solver

This repository contains the implementation of a general Runge-Kutta solver for differential equations for initial value problems (IVPs). This solver offers the option of transferring a control variable to the system, which is why it is suitable for control engineering applications. A specific butcher table is passed when the solver is instantiated, which is why different explicit solution variants can be implemented quickly.

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System Definition

The differential equation must be in the following form in order to be solved with this solver. In c++ code, for example, this can be formulated as a lambda expression.

$$\dot{x}(t) = f(t, x(t), u(t))$$ $$x(t_0) = x_0, \quad u(\cdot)$$

With:

$$ \begin{aligned} t &:~~ \text{Time, } t \in [t_0, b] \\ x &:~~ \text{System state, consisting of } n \text{ state variables } x \in \textit{Y} \subseteq \mathbb{R}^n \\ u &:~~ \text{Control input, consisting of } m \text{ control variables }u \in \textit{U} \subseteq \mathbb{R}^m \end{aligned} $$

Generalized Runge-Kutta Methods

Generalized Runge-Kutta methods are represented using the following formulas:

$$x_{k+1} = x_k + h \cdot \sum_{j=1}^{s} b_j \cdot k_j$$ $$k_j = f\left(t_k + h \cdot c_j, x_k + h \cdot \sum_{l=1}^{j-1} a_{jl} \cdot k_l, u_k\right)$$

Butcher Tableau

These methods are defined using a Butcher Tableau:

The attached test code generates a similar image to the one on the Wikipedia page. https://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods