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Journal of Computational Mathematics

Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps
Julian Koellermeier1; Giovanni Samaey2
1 Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, Netherlands ; Groningen Cognitive Systems and Materials Center, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, Netherlands ; Department of Computer Science, KU Leuven, 3001 Leuven, Belgium
2 Department of Computer Science, KU Leuven, 3001 Leuven, Belgium
The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 295-325.

Stiff hyperbolic balance laws exhibit large spectral gaps, especially if the relaxation term significantly varies in space. Using examples from rarefied gases and the general form of the underlying balance law model, we perform a detailed spectral analysis of the semi-discrete model that reveals the spectral gaps. Based on that, we show the inefficiency of standard time integration schemes expressed by a severe restriction of the CFL number. We then develop the first spatially adaptive projective integration schemes to overcome the prohibitive time step constraints of standard time integration schemes. The new schemes use different time integration methods in different parts of the computational domain, determined by the spatially varying value of the relaxation time. We use our analytical results to derive accurate stability bounds for the involved parameters and show that the severe time step constraint can be overcome. The new adaptive schemes show good accuracy in a numerical test case and can obtain a large speedup with respect to standard schemes.

Published online:
DOI: 10.5802/smai-jcm.88
Classification: 76P05, 35Q20, 35B40, 34E13, 35L02
Keywords: Projective integration, spatial adaptivity, hyperbolic balance law, moment equations
Julian Koellermeier 1; Giovanni Samaey 2

1 Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, Netherlands ; Groningen Cognitive Systems and Materials Center, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, Netherlands ; Department of Computer Science, KU Leuven, 3001 Leuven, Belgium
2 Department of Computer Science, KU Leuven, 3001 Leuven, Belgium
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     title = {Spatially {Adaptive} {Projective} {Integration} {Schemes} {For} {Stiff} {Hyperbolic} {Balance} {Laws} {With} {Spectral} {Gaps}},
     journal = {The SMAI Journal of computational mathematics},
     pages = {295--325},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
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     year = {2022},
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Julian Koellermeier; Giovanni Samaey. Spatially Adaptive Projective Integration Schemes For Stiff Hyperbolic Balance Laws With Spectral Gaps. The SMAI Journal of computational mathematics, Volume 8 (2022), pp. 295-325. doi : 10.5802/smai-jcm.88. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.88/

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