Stringent error estimates for one-dimensional, space-dependent 2×2 relaxation systems

Amadori, Debora; Gosse, Laurent

doi:10.1016/j.anihpc.2015.01.001

Amadori, Debora ; Gosse, Laurent

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 3, pp. 621-654.

Résumé

Sharp and local $L^{1}$ a posteriori error estimates are established for so-called “well-balanced” BV (hence possibly discontinuous) numerical approximations of $2 \times 2$ space-dependent Jin–Xin relaxation systems under sub-characteristic condition. According to the strength of the relaxation process, one can distinguish between two complementary regimes: 1) a weak relaxation, where local $L^{1}$ errors are shown to be of first order in Δx and uniform in time, 2) a strong relaxation, where numerical solutions are kept close to entropy solutions of the reduced scalar conservation law, and for which Kuznetsov's theory indicates a behavior of the $L^{1}$ error in $t \cdot \sqrt{Δ x}$ . The uniformly first-order accuracy in weak relaxation regime is obtained by carefully studying interaction patterns and building up a seemingly original variant of Bressan–Liu–Yang's functional, able to handle BV solutions of arbitrary size for these particular inhomogeneous systems. The complementary estimate in strong relaxation regime is proven by means of a suitable extension of methods based on entropy dissipation for space-dependent problems. Preliminary numerical illustrations are provided.

Zbl

DOI : 10.1016/j.anihpc.2015.01.001

Classification : 35L60, 65M06
Mots clés : Bressan–Liu–Yang functional, Entropy dissipation, Kuznetsov's method, $ {L}^{1}$ error estimate, Space-dependent relaxation model

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     author = {Amadori, Debora and Gosse, Laurent},
     title = {Stringent error estimates for one-dimensional, space-dependent 2{\texttimes}2 relaxation systems},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {621--654},
     publisher = {Elsevier},
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Amadori, Debora; Gosse, Laurent. Stringent error estimates for one-dimensional, space-dependent 2×2 relaxation systems. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 3, pp. 621-654. doi : 10.1016/j.anihpc.2015.01.001. http://archive.numdam.org/articles/10.1016/j.anihpc.2015.01.001/

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