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Illustration using the forward Euler, backward Euler, and Crank-Nicolson methods |
m Connect the description with IMEX schemes. |
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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 analytical solution exists. Then one uses [[root-finding algorithm]]s, such as [[Newton's method]], to find the numerical solution.
;Crank Nicoloson method
With the [[Crank-Nicolson method]]
:<math>\frac{y_{k+1}-y_k}{\Delta t} = -\frac{1}{2}y_{k+1}^2 -\frac{1}{2}y_{k}^2</math>
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: <math>y_{k+1}+\frac{1}{2}\Delta t y_{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>.
Crank Nicolson can be viewed as a form of more general [[Explicit and implicit methods|IMEX (''Im''plicit-''Ex''plicit) schemes]].
;Forward-Backward Euler method:
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