Explicit methods - Executable Engineering: Interactive Numerical Methods in Python

Explicit solver methods take information up to the current solution $y_i$ in order to directly predict the next step $y_{i+1}$ .

Runge-Kutta Methods¶

The Runge-Kutta methods are a class of time-stepping techniques where the next time step is calculated from the current time step and an estimate of its slope (rate of change). Increased accuracy is achieved through approximating higher-order derivatives to improve our time step. Since we don’t generally have analytical higher derivatives, we sample the rate function $f(t, y)$ at different positions across the step to approximate them.

Since we are time stepping, our independent variable is $t$ , and discretizing in time generates discrete times $t_i$ and their corresponding solutions $y(t=t_i) = y_i$ .

For first-order (in time) equations, which include parabolic PDEs:

\frac{\mathrm{d}y}{\mathrm{d}t} = y'_i = f(t_i, y_i)

(1)

Consider the Taylor expansion:

\begin{aligned} y_{i+1} &= y_i + y'_i h + \frac{y''_i}{2}h^2 + \frac{y'''_i}{6}h^3 + \cdots \\ &= y_i + f(t_i,y_i) h + \frac{f'(t_i,y_i)}{2}h^2 + \frac{f''(t_i,y_i)}{6}h^3 + \cdots \\ &\approx y_i + h \sum_{n=1}^s a_n k_n \end{aligned}

(2)

where the last line is the general form of the Runge-Kutta methods. The weights $a_n$ are constants and $k_n$ are the slope function evaluated at different positions in the interval $[t_i, t_{i+1}]$ . The goal is to match:

\sum_{n=1}^s a_n k_n \approx f(t_i,y_i) + \frac{f'(t_i,y_i)}{2}h + \frac{f''(t_i,y_i)}{6}h^2 + \cdots

(3)

as closely as possible with a series truncated in $s$ terms (called stages). Let’s build up to the general form step by step.