Glossary

Explicit and implicit methods

The distinction between computing the new step directly or by solving an equation that contains it.

Definition

An explicit method gives $y_{n+1}$ by a closed formula in already-known values, like explicit Euler $y_{n+1}=y_n+hf(t_n,y_n)$ . An implicit one has $y_{n+1}$ on both sides, like implicit Euler $y_{n+1}=y_n+hf(t_{n+1},y_{n+1})$ , and requires solving an equation at every step.

How it is used

You pay for the implicit step (solving an equation) in exchange for stability: in stiff problems, that trade is very profitable.