Explicit and implicit methods: Difference between revisions

Revision as of 01:48, 11 January 2017

Revision as of 12:48, 25 May 2018

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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.

Since the implicit method cannot be carried out for each kind of differential operator, it is sometimes advisable to make use of the so called operator splitting method, which means that the differential operator is rewritten as the sum of two complimentalcomplementary operators

:<math>Y(t+\Delta t) = F(Y(t+\Delta t))+G(Y(t)),\,</math>

while one is treated explicitly and the other implicitly.