1. Implicit and Explicit Runge-Kutta methods
Numerics of ODEs
Implicit and Explicit
Runge-Kutta methods

3. Runge–Kutta methods
The matrix [aij] is called the Runge–Kutta matrix, while the bi and ci are known as the weights and the nodes. These data are usually arranged in a mnemonic device, known as a Butcher tableau.

4. Gauss–Legendre method of order 4 (implicit)

5. Explicit Runge–Kutta methods

12. Classical Runge-Kutta: RK4 = Simpson’s rule generalized

16. RKF45 used by Maple
E. Fehlberg [Computing 6 (1970), 61–71] proposed and developed error control by using two RK methods of different orders. The difference of the computed y-values at gives an error estimate to be used for step size control. Fehlberg discovered two RK formulas that together need only six function evaluations per step. RKF has become quite popular. For instance, Maple uses it (also for systems of ODEs). Erwin Fehlberg (1969). Low-order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. NASA Technical Report 315.

20. Runge–Kutta–Nyström Methods (RKN Methods)

23. Stiff problems and implicit solutions

24. Euler method with h =0.1 for a stiff ODE and exact solution

30. Stability of the backward Euler method for any h;
Stability of the Euler method for h=0.1, but instability for h=0.2.
Stability of RK for h=0.2, but instability for h=0.3.
Euler method with h = 0.18 is shown in the figure on the left. Stability for x>3 (approximately).