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Numerical smoothness and error analysis on WENO for the nonlinear conservation laws
Author(s) -
Sun Tong
Publication year - 2013
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.21781
Subject(s) - smoothness , conservation law , mathematics , nonlinear system , scalar (mathematics) , piecewise , partial differential equation , error analysis , finite element method , numerical analysis , propagation of uncertainty , mathematical optimization , mathematical analysis , algorithm , geometry , physics , quantum mechanics , thermodynamics
In this study, we give an a posteriori error analysis on the weighted essentially nonoscillatory schemes for the nonlinear scalar conservation laws. This analysis is based on the new concept of numerical smoothness, with some new error analysis mechanisms developed for the finite difference and finite volume discretizations. The local error estimate is of optimal order in space and time. The global error estimate grows linearly in time, because of the direct application of the L 1 ‐contraction between entropy solutions in the error propagation analysis. As a beginning, we only deal with smooth solutions in this article. Within the same error propagation framework, when we deal with piecewise smooth solutions later, we only need to work on estimating the local error where smoothness is lost. The smoothness indicators not only serve the purpose of local error estimation, but also serve as a monitor on both the possible numerical instability and the expected solution shapening. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013