Explicit and implicit methods: Difference between revisions

Revision as of 15:45, 11 November 2009

Revision as of 13:24, 23 November 2009

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with the initial condition <math>y(0)=1.</math> Consider a grid <math>t_k=a\frac{k}{n}</math> for 0&le;''k''&le;''n'', that is, the time step is <math>\Delta t=a/n,</math> and denote <math>y_k=y(t_k)</math> for each <math>k</math>. [[Discretization|Discretize]] this equation using the simplest explicit and implicit methods, which are the ''forward Euler'' and ''backward Euler '' methods (see [[numerical ordinary differential equations]]) and compare the obtained schemes.

;Forward Euler method:

The forward Euler method

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

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for each <math>k=0, 1, \dots, n.</math> This is an explicit formula for <math>y_{k+1}</math>.

;Backward Euler method:

With the backward Euler method

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