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| A numerical tour of wave propagation | |
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To approximate the 1st-order derivatives as accurate as possible, we express it in the following
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(32) |
where
. Substituting the
and
with (29) for
results in
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(33) |
Thus, taking first
terms means
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(34) |
In matrix form,
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(35) |
The above matrix equation is Vandermonde-like system:
,
. The Vandermonde matrix
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(36) |
in which
, has analytic solutions.
can be solved using the specific algorithms, see Bjorck (1996). And we obtain
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(37) |
The MATLAB code for solving the 2N-order finite difference coefficients is provided in the following.
In general, the stability of staggered-grid difference requires that
|
(38) |
Define
. Then, we have
In the 2nd-order case, numerical dispersion is limited when
|
(39) |
The 4th-order dispersion relation is:
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(40) |
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| A numerical tour of wave propagation | |
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2021-08-31