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Appendix C: Stability analysis for $ \tau $ domain extrapolation

For homogeneous velocity and regularly spaced grids, the stability condition is often studied by Von Neumann analysis (Trefethen, 1994). Consider $ 1$ -D second-order wave equation,

$\displaystyle u_{tt} = v^2 u_{zz} ,$ (34)

approximated by central difference in space and time

$\displaystyle \frac{u_i^{n+1} - 2 u_i^n + u_i^{n-1}}{\Delta t^2} = v^2 \frac{u_{i+1}^n - 2 u_i^n + u_{i-1}^n}{\Delta z^2}$ (35)

where superscripts indicate time index and subscripts indicate space index. Substitute ansatz $ u_i^n = U^n\exp\left(\mathrm{i} k z_j\right)$ into Equation H-2 yields

$\displaystyle U^2 + \left( 4 \frac{v^2\Delta t^2}{\Delta z^2} \sin^2\frac{k\Delta z}{2} - 2 \right) U + 1 = 0$ (36)

where amplification factor $ U$ characterizes the growth of the numerical solution during iteration. Stability of the numerical solution requires $ \lvert U\rvert \leq 1$ for all wavenumbers $ k$ . It can be shown that in order for the quadratic equation

$\displaystyle U^2 + \beta U + 1 = 0 <tex2html_comment_mark>$ (37)

to have bounded roots $ \lvert U\rvert \leq 1$ , it is necessary that $ \lvert\beta\rvert \leq 2$ . This is equivalent to

$\displaystyle 0 \leq \frac{v^2\Delta t^2}{\Delta z^2} \sin^2\frac{k\Delta z}{2} \leq 1 \quad \forall \; k$ (38)

thus yields the well-known CFL condition

$\displaystyle \Delta t \leq \frac{\Delta x}{v} .$ (39)

In $ \tau $ domain, the $ 1$ -D problem can be extracted from the vertical component of Equation 19

$\displaystyle u_{tt} = \frac{v^2}{v_m^2} u_{\tau\tau} .$ (40)

Following the same analysis for H-2, we can obtain $ \tau $ domain CFL condition

$\displaystyle \Delta t \leq \frac{v_m}{v} \Delta \tau$ (41)

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2013-04-02