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), 707-724

MSC:
Primary 65M10; Secondary 65N10

DOI:
https://doi.org/10.1090/S0025-5718-1978-0492034-2

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:

*N*is the null space of

*K*and

*E*is the "negative" eigenspace of

*A*, then the system of partial differential equations is well-posed 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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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1978-0492034-2

Article copyright:
© Copyright 1978
American Mathematical Society