A necessary condition for the stability of a difference approximation to a hyperbolic system of partial differential equations
Author:
Anne M. Burns
Journal:
Math. Comp. 32 (1978), 707724
MSC:
Primary 65M10; Secondary 65N10
MathSciNet review:
492034
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Abstract: We are interested in the boundary conditions for a difference approximation to a hyperbolic system of partial differential equations , , in the quarter plane , . We consider approximations of the type: If N is the null space of K and E is the "negative" eigenspace of A, then the system of partial differential equations is wellposed if and only if and Rank the number of negative eigenvalues of A. In direct analogy to this, we prove that for a difference scheme of the above type with , and null space of , a necessary condition for stability is . If, in addition, a condition proven by S. J. Osher to be sufficient for stability is not satisfied, then Rank the number of negative eigenvalues of A is also necessary for stability. We then generalize this result to the case , . Together these conditions imply that "extrapolation" on "negative" eigenvectors leads to instability; "extrapolation" on "positive" eigenvectors is "almost necessary. "Extrapolation" on "positive" eigenvectors and not on "negative" eigenvectors is sufficient for stability.
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DOI:
http://dx.doi.org/10.1090/S00255718197804920342
PII:
S 00255718(1978)04920342
Article copyright:
© Copyright 1978
American Mathematical Society
