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Stability of Discrete Shocks for Difference Approximations to Systems of Conservation Laws

Daniel Michelson
SIAM Journal on Numerical Analysis
Vol. 40, No. 3 (2003), pp. 820-871
Stable URL: http://www.jstor.org/stable/4100905
Page Count: 52
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Stability of Discrete Shocks for Difference Approximations to Systems of Conservation Laws
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Abstract

The asymptotic stability of weak discrete stationary shocks for systems of conservation laws in one space dimension is proved. The difference approximation should be conservative, dissipative, and kth order accurate in space with odd k. The problem is considered in a finite interval $\vert x\vert \leq \ell$ with appropriate boundary conditions, where ℓ is large compared with the width of the shock layer $\varepsilon^{-1} = \vert u_{R} - u_{L}\vert^{-1/k}$. The proof is based on the assumption that the corresponding continuous shocks for the scalar problem $u_{t} + uu_{x} = -(i\partial_{x})^{k+1}u$ are stable. The latter is known to be true for k = 1 and k = 3.

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