Instability of difference models for hyperbolic initial boundary value problems.

*(English)*Zbl 0575.65095A theory of instability is presented for finite difference models of linear hyperbolic partial differential equations in one space dimension with a boundary. According to this theory, instability is caused by spurious radiation of wave energy from the boundary at a numerical group velocity \(C\geq 0\). To make this point of view precise, we first develop a rigorous description of group velocity for difference schemes and of reflection of waves at boundaries. From these results we then obtain lower bounds for growth rates of unstable finite difference solution operators in \(l^ 2\) norms, which extend earlier results due to Osher and to Gustafsson, Kreiss, and Sundström. In particular we investigate \(l^ 2\)-instability with respect to both initial and boundary data, and show how they are affected by (a) finite versus infinite reflection coefficients and (b) wave radiation with \(C=0\) versus \(C>0\).

##### MSC:

65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |

35L50 | Initial-boundary value problems for first-order hyperbolic systems |

##### Keywords:

instability; finite difference models; numerical group velocity; reflection coefficients; wave radiation
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\textit{L. N. Trefethen}, Commun. Pure Appl. Math. 37, 329--367 (1984; Zbl 0575.65095)

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