In [[applied mathematics]] and applications, '''explicit and implicit methods''' are approaches for mathematical [[computer simulation|simulatingsimulation]] of[[physics|physical]] [[process]]es, or toin putother it mathematically,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 atafter nextone instanceinterval inof time usingfrom 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 also be much harder to implement. TheImplicit reasonmethods oneare usesused implicitbecause methodsmany isproblems thatwhich forarise greatin manyreal problemslife (calledare [[stiff problem]]s), usingfor which the use of an explicit method requires extremelyimpracticably small time steps <math>\Delta t</math> forif the error ofin the methodresult tois not explodeto tend to infinity (see [[numerical stability]]). ItFor a given error on the result, 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. As such, theThe choice between whichof method to use depends onupon the specific problem atto be handsolved.
==Illustration using the forward and backward Euler methods==
