Pointwise error estimates for scalar conservation laws with piecewise smooth solutions

被引:29
作者
Tadmor, E [1 ]
Tang, T
机构
[1] Univ Calif Los Angeles, Dept Math, Los Angeles, CA 90095 USA
[2] Simon Fraser Univ, Dept Math, Vancouver, BC V5A 1S6, Canada
关键词
conservation laws; error estimates; viscosity approximation; optimal convergence rate; transport inequality;
D O I
10.1137/S0036142998333997
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We introduce a new approach to obtain sharp pointwise error estimates for viscosity approximation (and, in fact, more general approximations) to scalar conservation laws with piecewise smooth solutions. To this end, we derive a transport inequality for an appropriately weighted error function. The key ingredient in our approach is a one-sided interpolation inequality between classical L-1 error estimates and Lip(+) stability bounds. The one-sided interpolation, interesting for its own sake, enables us to convert a global L-1 result into a (nonoptimal) local estimate. This, in turn, provides the necessary bounds on the coefficients of the above-mentioned transport inequality. Estimates on the weighted error then follow from the maximum principle, and a bootstrap argument yields optimal pointwise error bound for the viscosity approximation. Unlike previous works in this direction, our method can deal with finitely many waves with possible collisions. Moreover, in our approach one does not follow the characteristics but instead makes use of the energy method, and hence this approach could be extended to other types of approximate solutions.
引用
收藏
页码:1739 / 1758
页数:20
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