Study guide: Truncation Error Analysis

Processing math: 100%

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Extension to quadratic damping

$\begin{equation} mu'' + \beta |u'|u' + s(u) =F(t)\tp \tag{41} \end{equation}$ Centered scheme:

$|u'|u'$ gives rise to a nonlinearity.

Linearization trick: use a geometric mean, $[|u'|u']^n \approx |[u']^{n-\half}|[u']^{n+\half}\tp$

Scheme: $\begin{equation} [mD_t D_t u]^n + \beta |[D_{t} u]^{n-\half}|[D_t u]^{n+\half} + s(u^n)=F^n\tp \end{equation}$