A necessary condition for the stability of a difference approximation to a hyperbolic system of partial differential equations

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: \[ {u_j}(t + \Delta t) = \sum \limits _{k = - r}^p {{C_k}{u_{j + k}}(t),\quad j = 1,2, \ldots ,} \] \[ {u_j} + \sum \limits _{k = 1}^s {{\alpha _{jk}}{u_k}(t + \Delta t) = 0,\quad j = - r + 1, \ldots ,0.} \] 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.