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First‐order L 1 ‐convergence for relaxation approximations to conservation laws
Author(s) -
Teng Zhen–Huan
Publication year - 1998
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199808)51:8<857::aid-cpa1>3.0.co;2-4
Subject(s) - conservation law , mathematics , piecewise , classification of discontinuities , rate of convergence , upper and lower bounds , mathematical analysis , relaxation (psychology) , entropy (arrow of time) , approximation error , physics , thermodynamics , psychology , social psychology , channel (broadcasting) , electrical engineering , engineering
We derive a first‐order rate of L 1 ‐convergence for stiffrelaxation approximations to its equilibrium solutions, i.e.,piecewise smooth entropy solutions with finitely manydiscontinuities for scalar, convex conservation laws. The piecewisesmooth solutions include initial central rarefaction waves, initialshocks, possible spontaneous formation of shocks in a future time,and interactions of all these patterns. A rigorous analysis showsthat the relaxation approximations to approach the piecewise smoothentropy solutions have L 1 ‐error bound of O (ε|logε| + ε), where ε is the stiff relaxationcoefficient. The first‐order L 1 ‐convergence rate is animprovement on the $O(\sqrt{\varepsilon})$ error bound. If neithercentral rarefaction waves nor spontaneous shocks occur, the errorbound is improved to O (ε). © 1998 John Wiley & Sons, Inc.