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# Absorbing boundary condition for modeling

Let us consider the explicit finite-difference scheme for the full wave equation
 (8)
where p is the wavefield and f is the force. We can extrapolate the wavefield along t using the following explicit finite-difference scheme
 (9)
where and are the second order central finite-difference operators

Given the initial condition pt-1x,z and ptx,z, we can solve equation (9) to get the wavefield at time t+1, pt+1x,z from the wavefield at time t-1, pt-1x,z and the wavefield at time t,ptx,z, except for the wavefield at the boundaries px=X<<177>>min,zt+1,px=X<<179>>max,zt+1, px,z=Z<<181>>mint+1,px,z=Z<<183>>maxt+1.

Let us consider the wavefield on the boundary z=Zmax. There are only outgoing waves at z=Zmax, so the wavefield satisfies the downgoing wave equation, for which we can write its approximate equations:
 (10) (11)

For compatibility with the explicit finite-difference scheme at internal points, we apply the explicit finite-difference scheme for the boundaries using equation (10) and (11) and get
 (12) (13)
where is the first order backward finite-difference operator, is the first order forward finite-difference opertor:

and is the second order central finite-difference operator:

Assuming that the wavefield px,zk for is known, then we solve the internal equation (9) to get the wavefield for the internal points Xmin<x<Xmax, Zmin<z<Zmax at time t+1, px,zt+1 first. Then, the auxiliary wavefield can be solved by equation (13) since the wavefield of the boundary at time t, ptx,Z<<275>>max and ptx,Z<<276>>max-1 are known. Finally, we solve equation (12) to get the wavefield at the boundary ptx,z=Z<<278>>max. Figure 1 illustrates how the boundary conditions are solved.

 boundary Figure 1 solution at the boundary z=Zmax

The method of solving the wavefield at the other three boundaries z=Zmin, x=Xmin, and x=Xmax, is similar to that of boundary z=Zmax. The only difference is that the boundary condition equation is an upgoing wave equation at z=Zmin, leftgoing wave equation at x=Xmin, and right-going wave equation at x=Xmax.

According to Zhang and Wei (1998), this absorbing boundary condition is stable.

Next: Numerical Example Up: Guojian: Absorbing boundary condition Previous: One-way wave equation
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6/8/2002