Stability of Finite Difference Boundary Conditions
, by Jeff Thorson
What are the sufficient conditions on R to guarantee that the extrapolation
of the differential operator uz = -RU is stable? There are two possible conditions
of stability. The first (weak) condition is to reqire that at large enough z, the
energy in u(z) is bounded by the initial energy of u(0). This is equivalent to the
condition that all eigenvalues of R have a positive real part. It will be seen that in
the case of an operator R with absorbing boundaries u_ = ru+ it is sufficient that
r lie in the upper right-hand quadrant of the complex plane. The second stability
condition is stronger: the energy in u(z) must be less than or equal to the initial
energy u(0) for all z. For an operaotr with absorbing boundaries, the stronger
condition is satisfied when, additionally, the following is known to be true:
mu(S) <= e^(+mz)
where mu(s) is the condition number of eigenvectors of S of R, and m is the smallest
real part among the eigenvalues of R. Normal operators that satisfy weak stability
criterion automatically are strongly stable. But this is not true for the class of
non-normal operators, among which are R with absorbig side boundaries. Whether the
above equation is satisfied or not depends on the particular operator R at hand. There
are indications of what to do to enforce the above condition: one is to increase the
size of the attenuation factor epsilon. Interrelations between R and the second difference
operator T that may simplify the task of satisfying the strong stability condition are
given in the following sections of the paper.
STANFORD