In [[applied mathematics]], '''explicit and implicit methods''' are approaches for mathematical [[computer simulation|simulation]] of [[physics|physical]] [[process]]es, or in other words, they are [[numerical analysis|numerical methods]] for solving time-variable [[ordinary differential equation|ordinary]] and [[partial differential equation]]s.
Explicit methods calculate the state of a system afterat onethe intervalnext instance of time from the state of the system at the current time, while an implicit method finds it by solving an equation involving both the current system state and the future one. To put it in symbols, if <math>Y(t)</math> is the current system state and <math>Y(t+\Delta t)</math> is the state at the next instance in time (<math>\Delta t</math> is a small time step), then, for an explicit method
: <math>Y(t+\Delta t) = F(Y(t))\,</math>
while for an implicit method one solves an equation
Line 7:
to find <math>Y(t+\Delta t).</math>
It is clear that implicit methods require an extra computation (solving the above equation), and they can be much harder to implement. Implicit methods are used because many problems which arisearising in real life are [[stiff problem|stiff]]s, for which the use of an explicit method requires impracticablyimpractically small time steps <math>\Delta t</math> ifto keep the error in the result is not to tend to infinitybounded (see [[numerical stability]]). For asuch givenproblems, errorto onachieve thegiven resultaccuracy, it takes much less computational time to use an implicit method with larger time steps, even taking into account that one needs to solve an equation of the form (1) at each time step. TheThat choicesaid, ofwhether one should use an explicit or implicit method depends upon the problem to be solved.
==Illustration using the forward and backward Euler methods==
