Positivity preservation of implicit discretizations of the advection equation

Abstract
We analyze, from the viewpoint of positivity preservation, certain discretizations of a fundamental partial differential equation, the one-dimensional advection equation with periodic boundary condition. The full discretization is obtained by coupling a finite difference spatial semi-discretization (the second- and some higher-order centered difference schemes, or the Fourier spectral collocation method) with an arbitrary -method in time (including the forward and backward Euler methods, and a second-order method by choosing suitably). The full discretization generates a two-parameter family of circulant matrices , where each matrix entry is a rational function in and . Here, denotes the CFL number, being proportional to the ratio between the temporal and spatial discretization step sizes. The entrywise non-negativity of the matrix -- which is equivalent to the positivity preservation of the fully discrete scheme -- is investigated via discrete Fourier analysis and also by solving some low-order parametric linear recursions. We find that positivity preservation of the fully discrete system is impossible if the number of spatial grid points is even. However, it turns out that positivity preservation of the fully discrete system is recovered for \emph{odd} values of provided that and are chosen suitably. These results are interesting since the systems of ordinary differential equations obtained via the spatial semi-discretizations studied are \emph{not} positivity preserving.

Acknowledgements
The Application-domain Specific Highly Reliable IT Solutions project has been implemented with the support provided from the National Research, Development and Innovation Fund of Hungary, financed under the Thematic Excellence Programme TKP2020-NKA-06 (National Challenges Subprogramme) funding scheme. This work was supported by the King Abdullah University of Science and Technology (KAUST), 4700 Thuwal, 23955-6900, Saudi Arabia, and by the Leibniz Competition.

Publisher
arXiv

arXiv
2105.07403

Additional Links
https://arxiv.org/pdf/2105.07403.pdf

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