Systems of Equations - Executable Engineering: Interactive Numerical Methods in Python

In most engineering problems, we encounter a coupled system of $j$ first-order differential equations evolving with respect to a single independent variable $t$ :

\begin{aligned} \frac{\mathrm{d}y_1}{\mathrm{d}t} &= f_1(t, y_1, y_2, \dots, y_j) \\ \frac{\mathrm{d}y_2}{\mathrm{d}t} &= f_2(t, y_1, y_2, \dots, y_j) \\ &\;\;\vdots \\ \frac{\mathrm{d}y_j}{\mathrm{d}t} &= f_j(t, y_1, y_2, \dots, y_j) \end{aligned}

(1)

Happily, extending single-equation Runge-Kutta and multistep methods to coupled systems is completely straightforward! By assembling the state variables into a vector $\mathbf{y}(t) = [y_1(t), y_2(t), \dots, y_j(t)]^{\mathsf{T}}$ and the right-hand side rates into a vector function $\mathbf{f}(t, \mathbf{y})$ , the system takes the compact vector form $\frac{\mathrm{d}\mathbf{y}}{\mathrm{d}t} = \mathbf{f}(t, \mathbf{y})$ , and the $s$ -stage Runge-Kutta update becomes:

\mathbf{y}_{i+1} \approx \mathbf{y}_i + h \sum_{n=1}^{s} a_n \mathbf{k}_n

(2)

with vector stage slopes:

\mathbf{k}_n = \mathbf{f}\!\left(t_i + p_n h,\; \mathbf{y}_i + h \sum_{m=1}^{s} q_{nm} \mathbf{k}_m\right)

(3)