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Method Order Applicable to
[note 1]
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.


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