Explicit and implicit methods: Difference between revisions

Revision as of 00:17, 28 September 2018

Revision as of 00:17, 28 September 2018

Line 2:

'''Explicit and implicit methods''' are approaches used in [[numerical analysis]] for obtaining numerical approximations to the solutions of time-dependent [[ordinary differential equation|ordinary]] and [[partial differential equation]]s, as is required in [[computer simulation]]s of [[Process (science)|physical processes]].

'''Explicit methods''' calculate the state of a system at a later time from the state of the system at the current time, while '''implicit methods''' find a solution by solving an equation involving both the current state of the system and the later one. Mathematically, if <math>Y(t)</math> is the current system state and <math>Y(t+\Delta t)</math> is the state at the later titime (<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

: <math>G\Big(Y(t), Y(t+\Delta t)\Big)=0 \qquad (1)\,</math>

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 arising in practice are [[Stiff equation|stiff]], for which the use of an explicit method requires impractically small time steps <math>\Delta t</math> to keep the error in the result bounded (see [[numerical stability]]). For such problems, to achieve given accuracy, 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. That said, whether one should use an explicit or implicit method depends upon the problem to be solved.