Explicit methods

Explicit solver methods take information up to the current solution in order to predict the next step.

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 timestep. Since we don’t generally have that information, we can sample the function at different steps to approximate them. This is the basis for the Runge-Kutta family of methods.

Since we are time stepping, our independent variable will be \(t\), but since we will be descritizing in time, we will generate times \(t_i\) and their corresponding solutions, \(y(t=t_i)=y_i\).

For first order (in time) equations, which will include parabolic equations,

\[ \frac{dy}{dt} = y^\prime_i = f(t_i, y_i) \]

Consider the Taylor expansion,

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

where the last line is the general form of the Runge-Kutta methods. \(a_n\) are a set of constants and \(k_n\) are the function evaluated at different positions in the interval. The goal is clearly to match

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

in so far as possible with a series truncated in \(s\) terms (called stages).

Let’s build up to the general form step by step.