1 Week 04 : 16 and 18 September 2008

We now begin the definition and construction of Runge-Kutta methods. These one–step methods are essentially always stable, but designing Runge– Kutta methods which are consistent to high order can be difficult. This theory is presented in Sec. We have already seen several examples of Runge–Kutta methods: explicit and implicit Euler, the implicit midpoint rule, the explicit midpoint rule with Euler predictor, and so forth. Another example is the trapezoidal rule with Euler predictor: in Runge–Kutta notation it looks like k 1 = f (t n , u n) k 2 = f (t n + h, u n + hk 1) u n+1 = u n + h(1 2 k 1 + 1 2 k 2) The basic idea is to build a series of " stages " k i which approximate y ′ = f at various points, using samples of f from other stages, and approximate u n+1 − u n by a linear combination of the stages. Since each stage can involve the others, the right–hand side f ends up being evaluated in a complicated nonlinear way. The most famous example is the classical four-stage fourth–order scheme k 1 = f (t n , u n) k 2 = f (t n + 1 2 h, u n + h 1 2 k 1) k 3 = f (t n + 1 2 h, u n + h 1 2 k 2) k 4 = f (t n + h, u n + hk 3) u n+1 = u n + h(1 6 k 1 + 2 6 k 2 + 2 6 k 3 + 1 6 k 4). Implicit methods are harder to put in Runge-Kutta form: it's usually easiest to identify the stages as function evaluations by looking at the overall 1