Explicit and implicit methods: Difference between revisions

Revision as of 00:19, 9 January 2006

Revision as of 00:19, 9 January 2006

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with the initial condition <math>y(0)=1.</math> Consider a grid <math>t_k=ka/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 equationequations]]s). The forward Euler method

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

yields