Abstract
Modified Patankar (MP) schemes are conservative, linear implicit and unconditionally positivity preserving time-integration schemes constructed for production-destruction systems. For such schemes, a classical stability analysis does not yield any information about the performance. Recently, two different techniques have been proposed to investigate the properties of MP schemes. In Izgin et al. [ESAIM: M2AN, 56 (2022)], inspired from dynamical systems, the Lyapunov stability properties of such schemes have been investigated, while in Torlo et al. [Appl. Numer. Math., 182 (2022)] their oscillatory behaviour has been studied. In this work, we investigate the connection between the oscillatory behaviour and the Lyapunov stability and we prove that a condition on the Lyapunov stability function is necessary to avoid oscillations. We verify our theoretical result on several numerical tests.
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Acknowledgements
The author T. Izgin gratefully acknowledges the financial support by the Deutsche Forschungsgemeinschaft (DFG) through grant ME 1889/10-1 (project number 466355003). P. Öffner was supported by the Gutenberg Research College, JGU Mainz. D. Torlo (Sissa, Italy) was supported by a SISSA Mathematical Fellowship.
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Izgin, T., Öffner, P., Torlo, D. (2024). A Necessary Condition for Non-Oscillatory and Positivity Preserving Time-Integration Schemes. In: Parés, C., Castro, M.J., Morales de Luna, T., Muñoz-Ruiz, M.L. (eds) Hyperbolic Problems: Theory, Numerics, Applications. Volume II. HYP 2022. SEMA SIMAI Springer Series, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-031-55264-9_11
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