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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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
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 $ {u_t} = A{u_x}$, $ u(x,0) = F(x)$, $ Ku(0,t) = 0$ in the quarter plane $ x \geqslant 0$, $ t \geqslant 0$.

We consider approximations of the type:

$\displaystyle {u_j}(t + \Delta t) = \sum\limits_{k = - r}^p {{C_k}{u_{j + k}}(t),\quad j = 1,2, \ldots ,} $

$\displaystyle {u_j} + \sum\limits_{k = 1}^s {{\alpha _{jk}}{u_k}(t + \Delta t) = 0,\quad j = - r + 1, \ldots ,0.} $

If 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 $ K \cap E = \left\{ 0 \right\}$ and Rank $ K = $ the number of negative eigenvalues of A.

In direct analogy to this, we prove that for a difference scheme of the above type with $ r = p = 1$, $ K\prime = I + \Sigma _{k = 1}^s\;{\alpha _k}$ and $ N\prime = $ null space of $ K\prime$, a necessary condition for stability is $ N\prime \cap E = \left\{ 0 \right\}$. If, in addition, a condition proven by S. J. Osher to be sufficient for stability is not satisfied, then Rank $ K = $ the number of negative eigenvalues of A is also necessary for stability. We then generalize this result to the case $ r > 1$, $ p > 1$.

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: http://dx.doi.org/10.1090/S0025-5718-1978-0492034-2
PII: S 0025-5718(1978)0492034-2
Article copyright: © Copyright 1978 American Mathematical Society