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Existence and stability of stationary profiles of the lw scheme
Author(s) -
Smyrlis Yiorgos Sokratis
Publication year - 1990
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/cpa.3160430405
Subject(s) - mathematics , conservation law , monotone polygon , norm (philosophy) , scalar (mathematics) , scheme (mathematics) , viscosity solution , mathematical analysis , stability (learning theory) , law , geometry , computer science , machine learning , political science
Abstract In this paper we study the behavior of difference schemes approximating solutions with shocks of scalar conservation lawsWhen a difference scheme introduces artificial numerical diffusion, for example the Lax‐Friedrichs scheme, we experience smearing of the shocks, whereas when a scheme introduces numerical dispersion, for example the Lax‐Wendroff scheme, we experience oscillations which decay exponentially fast on both sides of the shock. In his dissertation. Gray Jennings studied approximation by monotone schemes. These contain artificial viscosity and are first‐order accurate; they are known to be contractive in the sense of any l p norm. Jennings showed existence and l 1 stability of traveling discrete smeared shocks for such schemes. Here we study similar questions for the Lax‐Wendroff scheme without artificial viscosity; this is a nonmonotone, second‐order accurate scheme. We prove existence of a one‐parameter family of stationary profiles. We also prove stability of these profiles for small perturbations in the sense of a suitably weighted l 2 norm. The proof relies on studying the linearized Lax‐Wendroff scheme.