Explicit and implicit methods: Difference between revisions

Revision as of 11:36, 23 June 2009

Revision as of 19:40, 18 July 2009

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

<strong>'''Explicit methods</strong>''' calculate the state of a system at a later time from the state of the system at the current time, while <strong>'''implicit methods</strong>''' 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 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

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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 real life are [[Stiff_equationStiff 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.

==Illustration using the forward and backward Euler methods==

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: <math>y_{k+1}=\frac{-1+\sqrt{1+4\Delta t y_k}}{2 \Delta t}. \quad \quad (4)</math>

In the vast majority of cases, the equation to be solved when using an implicit scheme is much more complicated than a quadratic equation, and no exact solution exists. Then one uses [[root-finding algorithm|root-finding algorithms]]s, such as [[Newton's method]].

==See also==

* [[Courant–Friedrichs–Lewy condition]]

{{DEFAULTSORT:Explicit And Implicit Methods}}

[[Category:Numerical differential equations]]