Semi implicit method
- Simplest semi-implicit method: Semi-implicit Euler / Verlet
- General case
-
- \(v^{k+1}=v^{k}+h\,\mathcal{F}_v(p^k,v^k,t^k)\)
- \(p^{k+1}=p^{k}+h\,\mathcal{F}_p(p^k,v^{k+1},t^k)\)
Application to classical mechanical cases
- \(p'(t)=v(t), m\,v'(t)=F(p,t)\)
- \(\Rightarrow \left\{\begin{array}{l}v^{k+1} = v^{k} + h\,F(p^k,t^k)/m \\ p^{k+1} = p^k + h\,v^{k+1}\end{array}\right.\)
- (+) Trivially easy to convert explicit Euler to semi-implicit Euler
- Expressed using positions only
-
- \(p^{k+1}= p^k + h(v^k+h\,F(p^k,t^k))\)
- \(p^{k+1}=p^k+h\left(\frac{p^k-p^{k-1}}{h}+h\,F(p^k,t^k)/m\right)\)
- \(\Rightarrow p^{k+1}= 2\, p^k - p^{k-1} + h^2\,F(p,t)/m\)
- 1st order accurate (like explicit/implicit Euler) in position and speed.