I am simulating acoustic wave equation in which absorbing boundary condition has to be applied. It is applied in two ways

Ist method is as mentioned in this paper. Boundary condition at bottom is (equation 9 in paper)
$$ P_{zt} + \frac{1}{v} P_{tt} - \frac{v}{2} P_{xx} $$

which is finite differenced as following (see appendix in above referred paper):

'Bottom boundary at k=K'

$$ D^{-}_z D_t^0 P_{i,K}^n + \frac{v}{2} D_t^+ D_{t}^{-} (P_{i,K}^n + P_{i,K-1}^n) - \frac{v}{4} D_x^+ D_{x}^{-} (P_{i,K-1}^{n+1} + P_{i,K}^{n-1}) = 0 $$

here $D$ represent the derivative; subscripts x,z and t represent the spatial directions (x and z) and time (t); superscripts +,- and 0 represent the forward, backward, and centered approximations for respective derivatives; and

(Notation used above is different from the paper. I am writing FD equations the way it is written generally.)

My question is about the last term.

What advantage we can get by writing last term in the form $ \frac{v}{4} D_x^+ D_{x}^{-} (P_{i,K-1}^{n+1} + P_{i,K+1}^{n-1}) $ instead of writing like $ \frac{v}{4} D_x^+ D_{x}^{-} (P_{i,K-1}^{n} + P_{i,K}^{n}) $. Both are centered at $ n\Delta t,(K-\frac{1}{2})\Delta z $.

Is it correct?

2nd way is the way defined in the paper "Boundary condition for the numerical solution of wave propagation problem , By Albert C reynolds, 1978, Geophysics(vol.43,No 6)" (I couldn't find its open access) The final equation is $$ (\frac{1}{c}\frac{\partial}{\partial t} + \frac{\partial}{\partial x}) (\frac{p}{c}\frac{\partial}{\partial t} + \frac{\partial}{\partial x}) =0 $$

I believe the author used the first part and written its finite differenced form as $$ u^{j+1}_{m,N+1} = u^j_{m,N+1} + u^j_{m,N} - u^{j-1}_{m,N} - \frac{c \Delta t}{\Delta x} (u^j_{m,N+1} - u^j_{m,N} - (u^{j-1}_{m,N} - u^{j-1}_{m,N-1})) $$ I interpreted above form as $$ D_t^+ u^j_{m,N+1} -D_t^- u^j_{m,N} = D_x^+ u^j_{m,N} - D^-_xu^{j-1}_{m,N} $$

here $$ u_{m,n}^j = u^{\Delta t*j}_{\Delta x *m, \Delta z* n} = u^t_{x,z} $$

here I could not find any cenerting in time or space. Also I found it uses difference insted of averaging...!

How it was formulated?


The spatial indices are $n$ and $k$ (see figure 3 in the paper). $i$ (called $j$ in the paper) seems to be the temporal index.

With these definitions, the term about which you inquire seems to be a temporal averaging, not a spatial one.

  • $\begingroup$ Erik regarding notation I have edited my question now. You are right about the temporal averaging. My question is why to incorporate that? If it is required then why it is not done in other terms as well. $\endgroup$
    – Amartya
    Aug 20 '16 at 12:37
  • $\begingroup$ Appendix 3 of the paper uses the indices $P^n_{j+1}$ etc., whereas you write $P^{n+1}_i$. There are two differences -- first, you replaced $j$ by $i$ without saying so, and second, the $+1$ is on the wrong index (upstairs instead of downstairs). I edited your question, but this edit was not accepted. $\endgroup$ Aug 21 '16 at 13:51
  • $\begingroup$ Yes Erik, first I forgot to mention the modified notation however later I modified the question accordingly and I mention that in my comment. I modified the notation because in modern notation $P^t_{x,z}$ subscripts are spatial index and superscript is time. It helps reader to understand it with ease. You modified the question according to the paper notation $P^x_{t,z}$ so I decline the edit. I appreciate the time you gave. I have modified the question see If you have any comment on that. $\endgroup$
    – Amartya
    Aug 22 '16 at 6:50

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