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Effective boundary for regular grid finite difference

Assume $ 2N$ -th order finite difference scheme is applied. The Laplacian operator is specified by

\begin{displaymath}\begin{array}{rl} \nabla^2 p^{k}&=\partial_{xx}p^{k}+\partial... ...frac{1}{\Delta x^2}\sum_{i=-N}^Nc_i p^k[ix+i][iz]\\ \end{array}\end{displaymath} (6)

where $ c_i$ is given by Table 1, see a detailed derivation in Fornberg (1988). The Laplacian operator has $ x$ and $ z$ with same finite difference structure. For $ x$ dimension only, the second derivative of order $ 2N$ requires at least $ N$ points in the boundary zone, as illustrated by Figure 2. In 2-D case, the required boundary zone has been plotted in Figure 3a. Note that four corners in $ B_1B_2B_3B_4$ in Figure 1 are not needed. This is exactly the boundary saving scheme proposed by Dussaud et al. (2008).


Table 1: Finite difference coefficients for regular grid (Order-$ 2N$ )
$ i$ -4 -3 -2 -1 0 1 2 3 4
$ N=1$       1 -2 1      
$ N=2$     -1/12 4/3 -5/2 4/3 -1/12    
$ N=3$   1/90 -3/20 3/2 -49/18 3/2 -3/20 1/90  
$ N=4$ -1/560 8/315 -1/5 8/5 -205/72 8/5 -1/5 8/315 -1/560

Keep in mind that we only need to guarantee the correctness of the wavefield in the original model zone $ A_1A_2A_3A_4$ . However, the saved wavefield in $ A_1A_2A_3A_4\backslash B_1B_2B_3B_4$ is also correct. Is it possible to further shrink it to reduce number of points for saving? The answer is true. Our solution is: saving the inner $ N$ layers on each side neighboring the boundary $ A_1A_2A_3A_4\backslash D_1D_2D_3D_4$ , as shown in Figure 3b. We call it the effective boundary for regular finite difference scheme.

After $ nt$ steps of forward modeling, we begin our backward propagation with the last 2 wavefield snap $ p^{nt}$ and $ p^{nt-1}$ and saved effective boundaries in $ A_1A_2A_3A_4\backslash D_1D_2D_3D_4$ . At that moment, the wavefield is correct for every grid point. (Of course, the correctness of the wavefield in $ A_1A_2A_3A_4$ is guaranteed.) At time $ k$ , we assume the wavefield in $ A_1A_2A_3A_4$ is correct. One step of backward propagation means $ A_1A_2A_3A_4$ is shrunk to $ D_1D_2D_3D_4$ . In other words, the wavefield in $ D_1D_2D_3D_4$ is correctly reconstructed. Then we load the saved effective boundary of time $ k$ to overwrite the area $ A_1A_2A_3A_4\backslash D_1D_2D_3D_4$ . Again, all points of the wavefield in $ A_1A_2A_3A_4$ are correct. We repeat this overwriting and computing process from one time step to another ( $ k\rightarrow k-1$ ), in reverse time order. The wavefield in the boundary $ C_1C_2C_3C_4\backslash A_1A_2A_3A_4$ may be incorrect because the points here are neither saved nor correctly reconstructed from the previous step.

fig2
fig2
Figure 2. 1-D schematic plot of required points in regular grid for boundary saving. Computing the laplacian needs $ N$ points in the extended boundary zone, the rest $ N+1$ points in the inner model grid. $ N$ points is required for boundary saving.
[pdf] [png]

fig3
fig3
Figure 3. A 2-D sketch of required points for boundary saving for regular grid finite difference: (a) The scheme proposed by Dussaud et al. (2008) (red zone). (b) Proposed effective boundary saving scheme (gray zone).
[pdf] [png]

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2021-08-31