Numerical solution

1st order Explicit Euler

Naive numerical scheme: Approximation of the derivative

- \(\displaystyle \frac{u^{k+1}-u^k}{h}=\mathcal{F}(u^k,t^k)\)

- \(\Rightarrow u^{k+1} = u^{k} + h\,\mathcal{F}(u^k,t^k)\)

In the linear case

- \(u^{k+1}=\left(\mathrm{I}+h\,\mathrm{A}\right)\,u^{k}+h\,b^k\)

Pro : very easy to implement

Is \(u^{k}\) a good approximation of the true solution \(\tilde u(t^k)\) ?

For a single particle with \(p,v\) variables: \( \left\{ \begin{array}{l} v^{k+1}=v^k + h\, F(u^k,t^k)/m \\ p^{k+1}=p^k + h\, v^k \end{array} \right. \)