Explicit and implicit methods: Difference between revisions

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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} y_y^{2}_{k+1}^2 = y_k - \frac{1}{2}\Delta t y_{k}^2</math>

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>.