A Stencil-width Five Scheme

A simple explicit generalization of scheme (6.33) involves a wider-stencil centered approximation to the spatial derivative term:

$$\delta_{tt}u = \gamma^2\left(\alpha + (1-\alpha)\mu_{x\cdot}\right)\delta_{xx}u$$ (6.60)

This scheme involves a free parameter $\alpha$, and reduces to scheme (6.33) when $\alpha = 1$; it is nominally second-order accurate in both space and time. The form above may be expanded into the following recursion:

$$u_{l}^{n+1} = (2+\lambda^2(1-3\alpha))u_{l}^n +\lambda^2(2\alpha-... ...{\lambda^2}{2}\left(1-\alpha\right)\left(u_{l+2}^n+u_{l-2}^n\right)-u_{l}^{n-1}$$ (6.61)

This scheme clearly makes use of points two grid spacings removed from the update point, and thus has a stencil width of five. See Figure (a) for a representation of the ``footprint" corresponding to this scheme.

The characteristic equation for scheme (6.60) is easily obtained, again through the insertion of a test solution of the form $u_{l}^{n}=z^{n}e^{jl\beta h}$:

$$z+\left(-2+4\lambda^2\left(1-2(1-\alpha)p\right)p\right)+z^{-1} = 0$$ (6.62)

where the short-hand notation $p=\sin^2(\beta h/2)$ has been used. The condition that the roots of the characteristic equation are of unit magnitude is that

$$0\leq \lambda^2\left(1-2(1-\alpha)p\right)p\leq 1$$ (6.63)

for all values of $0\leq p \leq 1$. The left-hand inequality is satisfied for

$$\alpha\geq 1/2$$ (6.64)

Given the above restriction on $\alpha$, the right hand inequality yields a bound on $\lambda$:

$$\lambda \leq \sqrt{\frac{1}{2\alpha-1}}$$ (6.65)

Notice that as $\alpha$ increases, $\lambda$ is forced to decrease.