4.1.4 Implicit Methods

Much more stable!

- First order scheme (from DNGB): Insert

$$\left. \frac{d\mbox{$\bf y$}}{dt} \right\vert _{n+1} = \frac{\nabla \mbox{$\bf y$}_{n+1}}{\Delta t}+O[\Delta t]$$

in $d\mbox{$\bf y$}/dt=f[\mbox{$\bf y$}(t)]$ to find

$$\mbox{$\bf y$}_{n+1} = \mbox{$\bf y$}_{n}+\mbox{$\bf f$}_{n+1} \Delta t + O[(\Delta t)^{2}]$$

If $\mbox{$\bf f$}(\mbox{$\bf y$})$ is linear, $\mbox{$\bf f$}_{n+1}=\mbox{${\bf L}$} \cdot \mbox{$\bf y$}_{n+1}$:

$$\fbox{$ \displaystyle \mbox{$\bf y$}_{n+1}=[\mbox{${\bf I}$}-\mbox{${\bf L}$} \Delta t]^{-1} \cdot \mbox{$\bf y$}_{n} +O[(\Delta t)^{2}] $}$$

Stability:

$$\mbox{$\bf e$}_{n+1}=[\mbox{${\bf I}$}-\mbox{${\bf L}$} \Del... ...{$\bf e$}_{n} \equiv \mbox{${\bf G}$} \cdot \mbox{$\bf e$}_{n}$$

(a) Relaxation equation: $\mbox{${\bf G}$}=G=1/(1+\lambda \Delta t)$, thus $\vert g\vert<1$ for any $\lambda>0$.

(b) Harmonic oscillator:

$$\mbox{${\bf G}$} \equiv [\mbox{${\bf I}$}-\mbox{${\bf L}$} \... ...\vspace{-9pt}\\ -\omega_{0}^{2}\Delta t&1\end{array} \right)$}$$

with eigenvalues

$$g_{1,2}=\frac{1}{1+(\omega_{0} \Delta t)^{2}} [1 \pm i \omega_{0} \Delta t]$$

Thus

$$\vert g\vert^{2}=\frac{1}{1+(\omega_{0} \Delta t)^{2}} < 1 \;\;\; {\rm for}\;\;{\rm all}\;\; \Delta t$$

$\Longrightarrow$Method is always stable for relaxation equation and harmonic oscillator.

- Second order implicit scheme (from adding the DNGF formulae at $t_{n}$ and $t_{n+1}$, respectively):

$$\mbox{$\bf y$}_{n+1} = \mbox{$\bf y$}_{n} +\frac{\Delta t}{2}[\mbox{$\bf f$}_{n}+\mbox{$\bf f$}_{n+1}] +O[(\Delta t)^{3}]$$

If $\mbox{$\bf f$}_{n}=\mbox{${\bf L}$} \cdot \mbox{$\bf y$}_{n}$ etc.:

$$\fbox{$ \mbox{$\bf y$}_{n+1}=[\mbox{${\bf I}$}-\mbox{${\bf L}$} \... ...{${\bf L}$} \frac{\Delta t}{2}] \cdot \mbox{$\bf y$}_{n} +O[(\Delta t)^{3}] $ }$$

Always stable for relaxation equation and harmonic oscillator.