Explicit and implicit methods: Difference between revisions

Revision as of 18:19, 23 April 2019

Revision as of 22:02, 11 July 2019

Line 13:

:<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>).

for Time-Dependent Partial Differential Equations'', Appl Numer Math, vol. 25(2-3), 1997</ref>, <ref>L.Pareschi, G.Russo: ''Implicit-Explicit Runge-Kutta schemes for stiff systems of differential equations'', Recent Trends in Numerical Analysis, Vol. 3, 269-289, 2000</ref>).

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