The Predictor-Corrector Method

This method is very similar to and often confused with the Runge-Kutta method. We consider substituting the trapezoidal rule for the estimate of the integral in (1.8) to obtain the equation

$$\bi{y}_{n+1} = \bi{y}_n - {\mathchoice{{\textstyle{\frac12}... ...[\bi{f}(\bi{y}_{n+1}, t_{n+1}) + \bi{f}(\bi{y}_n,t_n)\right].$$ (1.38)

Unfortunately the presence of $\bi{y}_{n+1}$ on the right hand side makes a direct solution of (1.38) impossible except for special cases. A possible compromise is the following method

$$y'_{n+1} = y_n - \delta t f(y_n, t_n)$$ (1.39)

$$y_{n+1} = y_n - {\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{... ...riptscriptstyle{1/2}}}}\delta t \left[f(y'_{n+1}, t_{n+1} + f(y_n, t_n)\right].$$ (1.40)

This method consists of a guess for $y_{n+1}$ based on the Euler method (the Prediction) followed by a correction using the trapezoidal rule to solve the integral equation. The accuracy and stability properties are identical to those of the Runge-Kutta method.