3. Implicit Runge-Kutta Methods

Runge-Kutta methods can be either an explicit or implicit methods depending on the functions used to calculate the stage values. We have seen in the previous chapter on explicit methods that they are straightforward to apply, however they are not always suitable for solving ODEs that are stiff. This is why we need to also consider implicit methods.

Recall that the general form of a Runge-Kutta method to solve a first-order ODE \(y'=f(t,y)\) is

\[\begin{split} \begin{align*} y_{n+1} &=y_n + h \sum_{i=1}^s b_i k_i,\\ k_i &=f(t_n +c_i h,y_n + h \sum_{j=1}^s a_{ij} k_j). \end{align*} \end{split}\]

Expanding out the stage value functions we have

\[\begin{split}\begin{align*} k_1 &= f(t_n + c_1 h, y_n + h (a_{11} k_1 + a_{12} k_2 + \cdots + a_{1s} k_s), \\ k_2 &= f(t_n + c_2 h, y_n + h (a_{21} k_1 + a_{22} k_2 + \cdots + a_{2s} k_s), \\ & \vdots \\ k_s &= f(t_n + c_1 h, y_n + h (a_{s1} k_1 + a_{s2} k_2 + \cdots + a_{ss} k_s). \end{align*}\end{split}\]

The value of \(k_1\) appears withing the function on the right-hand side of \(k_1\) and similar for the other stage values. This means these functions are implicit functions hence we have an implicit Runge-Kutta method. The Butcher tableau for an implicit method is

\[\begin{split} \begin{array}{c|cccc} c_1 & a_{11} & a_{12} & \cdots & a_{1s} \\ c_2 & a_{21} & a_{22} & \cdots & a_{2s} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ c_s & a_{s1} & a_{s2} & \cdots & a_{ss} \\ \hline & b_1 & b_2 & \cdots & b_s \end{array} \end{split}\]

Note that the \(A\) matrix in the upper right-hand region of an implicit method can have non-zero elements in them main diagonal and upper triangular region whereas the \(A\) matrix for an explicit method has non-zero elements in the lower triangular region only.