Geodesic
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

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Sharp and local L1 a posteriori error estimates are established for so-called “well-balanced” BV (hence possibly discontinuous) numerical approximations of 2×2 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 L1 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 L1 error in t⋅Δx. 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.

DOI : 10.1016/j.anihpc.2015.01.001
Classification : 35L60, 65M06
Keywords: 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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     doi = {10.1016/j.anihpc.2015.01.001},
     zbl = {1339.35170},
     language = {en},
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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

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