User:Fuse809/sandbox

Method	Order	Applicable to	Implicit/ explicit?	Adaptive?	Notes
Backward Euler method	1st	All initial value problems (IVPs).	Implicit.	No	Uses the definition of a derivative.
Euler method	1st	All IVPs.	Explicit.	No	Uses the definition of a derivative.
Heun's method	2nd	All IVPs.	Explicit.	No	Also known as the explicit trapezoidal rule and modified Euler method.
Midpoint method	2nd	All IVPs.	Explicit.	No	Uses the definition of a derivative.
Runge-Kutta-Fehlberg method	Varies	All IVPs.	Explicit.	Yes	A type of Runge-Kutta method that uses an adaptive step size to achieve a desired level of accuracy.
Runge-Kutta methods	Varies.	All IVPs.	Both.	Varies	A family of methods of variable order, some of which are implicit while others are explicit. Some have an adaptive step size, but others do not. The most popular variant is likely the explicit non-adaptive 4th order Runge-Kutta method (RK4).
Trapezoidal rule	2nd	All IVPs.	Implicit.	No	ODE counterpart to the trapezoidal rule for numerical definite integration.