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• W2090820002 abstract "The orbital stability of possibly large semidiscrete shock waves is considered. These waves are traveling wave solutions of discrete in space and continuous in time systems of conservation laws, which constitute a class of lattice dynamical systems (LDSs). The underlying lattice $Delta x mathbb{Z}$ is by nature not invariant by change of frame. Thus semidiscrete shock waves cannot really be transformed into stationary waves, unlike other kinds of approximate shock waves (e.g., viscous or relaxation shocks). This implies that the linearization of the LDS about a given semidiscrete shock wave yields a nonautonomous linear LDS, which cannot be tackled by means of Laplace transform in time. However, viewing the LDS as a finite-difference PDE and performing afterall the change of frame, the profile becomes a stationary solution of the transformed equation. Then, linearizing about the profile, we get an evolution finite-difference PDE in which the spatial operator L, a delayed and advanced differential operator, plays a crucial role in our stability analysis. In particular, we point out an integral formula relating the Green's function of the linearized LDS to the Green's function $G_{lambda}$ of $(lambda-L)$. Specializing to the upwind scheme, we take advantage of the material introduced in an earlier work [S. Benzoni-Gavage, J. Dynam. Differential Equations, 14 (2002), pp. 613--674], in particular of an Evans function, to decompose the Green's function similarly as Zumbrun et al. did for other approximate shock waves. This decomposition relies on explicit representations of the projections involved in the exponential dichotomies and their extensions through the gap lemma. It enables us in turn to prove the orbital stability of the wave, provided that the Evans function D does not vanish in the right half-plane but on the discrete set $2ipisigma mathbb{Z}$ ($sigma$ being the speed of the wave) and that D has a simple root at 0. Additionally, we show that this spectral stability condition is satisfied at least for (extreme) weak shocks. Our spectral stability condition, and more specifically the $D'(0)neq 0$ part, appears to be a relaxed version of the requirement of Chow, Mallet-Paret, and Shen [J. Differential Equations, 149 (1998), pp. 248--291], which we show to be too strong for traveling waves in conservative LDSs." @default.
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• W2090820002 date "2003-01-01" @default.
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• W2090820002 title "Nonlinear Stability of Semidiscrete Shock Waves" @default.
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• W2090820002 doi "https://doi.org/10.1137/s0036141002418054" @default.
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