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In solving the wave equation by finite differencing, we should use central
finite-difference schemes because the wave equation and physical wave
phenomena are symmetric. And near a boundary with boundary conditions, we
should gradually drop the order of finite differencing for the derivative
perpendicular to the boundary.
The nth order central finite differencing for the second derivative is

| |
(1) |

For the 2nd order:
w0=-2.; w1=1.

For the 4th order:
w0=-5./2.; w1=4./3.; w2=-1./12.

For the 6th order:
w0=-2.72222; w1=1.50000; w2=-0.15000; w3=1.11111E-02.

For the 8th order:
w0=-2.84722; w1=1.60000; w2=-0.20000; w3=2.53968E-02; w4=-1.78571E-03.

For the 10th order^{}:
w0=-2.92722; w1=1.66667; w2=-0.238095; w3=3.96825E-02;
w4=-4.96031E-03; w5=3.17460E-04.

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Stanford Exploration Project

11/17/1997