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to find <math>Y(t+\Delta t).</math>
== Computation ==
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.
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:<math>Y(t+\Delta t) = F(Y(t+\Delta t))+G(Y(t)),\,</math>
while one is treated explicitly and the other implicitly.
For usual applications the implicit term is chosen to be linear while the explicit term can be nonlinear. This combination of the former method is called ''Implicit-Explicit Method'' (short IMEX,<ref>U.M. Ascher, S.J. Ruuth, R.J. Spiteri: ''[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.48.1525&rep=rep1&type=pdf Implicit-Explicit Runge-Kutta Methods for Time-Dependent Partial Differential Equations]'', Appl Numer Math, vol. 25(2-3), 1997</ref><ref>L.Pareschi, G.Russo: ''[https://www.researchgate.net/profile/Lorenzo_Pareschi/publication/230865813_Implicit-Explicit_Runge-Kutta_schemes_for_stiff_systems_of_differential_equations/links/0046352a03ba3ee92a000000.pdf Implicit-Explicit Runge-Kutta schemes for stiff systems of differential equations]'', Recent Trends in Numerical Analysis, Vol. 3, 269-289, 2000</ref><ref>Sebastiano Boscarino, Lorenzo Pareschi, and Giovanni Russo: ''Implicit-Explicit Methods for Evolutionary Partial Differential Equations'', SIAM, ISBN 978-1-61197-819-3 (2024).</ref>).
==Illustration using the forward and backward Euler methods==
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one finds the implicit equation
: <math>y_{k+1}+\frac{1}{2}{\Delta t}
for <math>y_{k+1}</math> (compare this with formula (3) where <math>y_{k+1}</math> was given explicitly rather than as an unknown in an equation). This can be numerically solved using [[root-finding algorithm]]s, such as [[Newton's method]], to obtain <math>y_{k+1}</math>.
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