Explicit and implicit methods: Difference between revisions

Revision as of 22:03, 11 July 2019

Revision as of 18:39, 27 December 2019

Line 23:

;Forward Euler method:

[[File:ForwardResult andof backwardapplying Eulerintegration methodschemes.png|thumb|350px|The result of applying thedifferent twointegration methods withto ode <math>a y'=-y^2, \; t\in[0, 5], \; y_0=1 </math> andwith <math>n\Delta t = 305/10</math>.]]

The forward [[Euler method]]

:<math>\left(\frac{dy}{dt}\right)_k \approx \frac{y_{k+1}-y_k}{\Delta t} = - y_k^2</math>

Line 42:

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 (An IMEX scheme):

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>

one finds the implicit equation

: <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>.

;Forward-Backward Euler method: