Explicit and implicit methods: Difference between revisions

Revision as of 22:17, 28 July 2021

Revision as of 14:41, 18 January 2022

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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 -Nicolson 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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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 IMEX (''Im''plicit-''Ex''plicit) schemes.

;Forward-Backward Euler method: