What distinguishes implicit from explicit time-stepping methods?

Implicit time-stepping methods are characterized by their consideration of future states in conjunction with the current state. This means that when solving equations, implicit methods involve the unknown values (future states) in the calculation, requiring the solution of a system of equations at each time step. This can introduce more complexity in terms of computational effort but also offers better stability properties, particularly for stiff equations.

In contrast, explicit methods do not incorporate future states into their calculations. Instead, they rely on previously computed (past) values to establish the state of the system at the next timestep. This typically allows for simpler computations but can lead to stability issues, limiting the size of the time step that can be chosen.

The distinction between these methods is critical in engineering computations, especially in time-dependent simulations, as one method may be more suitable than the other depending on the system dynamics being analyzed.