Taylor method

The Taylor method is a one-step method in numerics . It is a way of constructing higher order difference formulas using the Taylor expansion .

Derivation

Based on an initial value problem ( AWA) of the form: and the Taylor formula , the scalar case is considered. ${\displaystyle u'(t)=f(t,u(t)),u(t_{0})=u_{0}}$ ${\displaystyle d=1}$

${\displaystyle u(t)=\sum _{r=0}^{R}{\frac {h^{r}}{r!}}u^{(r)}(t-h)+{\frac {h^{(R+1)}}{(R+1)!}}u^{(R+1)}(\xi ),\xi \in [t-h,t]}$

Since the differential equation is sufficient, the following applies ${\displaystyle u}$ ${\displaystyle u'=f(t,u(t))}$

${\displaystyle u^{(r)}(t)={\begin{pmatrix}d\\dt\end{pmatrix}}^{r-1}f(t,u(t))=:f^{r-1}(t,u(t))}$

The -step Taylor procedures are then ${\displaystyle R}$

${\displaystyle y_{n}=y_{n-1}+h_{n}\sum _{r=1}^{R}{\frac {h_{n}^{r-1}}{r!}}f^{(r-1)}(t_{n-1},y_{n-1}),\ \ n\geq 1}$

The Taylor method has the consistency order (numerics) ${\displaystyle m=R}$

Numerical stability

We apply the test equation to the procedure: ${\displaystyle y_{n}=y_{n-1}+h\sum _{r=1}^{R}{\frac {h_{n}^{r-1}}{r!}}f^{(r-1)}(t_{n-1},y_{n-1}){\underset {u'=\lambda u}{=}}y_{n-1}+h\sum _{r=1}^{R}{\frac {h_{n}^{r-1}}{r!}}\lambda ^{(r)}y_{n-1}=y_{n-1}(1+\sum _{r=1}^{R}{\frac {h_{n}^{r}}{r!}}\lambda ^{(r)}}$

The gain factor is accordingly ${\displaystyle \omega =\sum _{r=0}^{R}{\frac {(\lambda h)^{r}}{r!}}}$

Individual evidence

1. ^ Rolf Rannacher: Numerics 1. Numerics of ordinary differential equations . Heidelberg 2017, p. 46 ff .