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Aliasing and two-dimensional well-balanced for drift-diffusion equations on square grids

Author: Laurent Gosse
Journal: Math. Comp. 89 (2020), 139-168
MSC (2010): Primary 35K15, 65M12, 76D05, 76R50
Published electronically: July 16, 2019
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Abstract: A notion of ``2D well-balanced'' for drift-diffusion is proposed. Exactness at steady-state, typical in 1D, is weakened by aliasing errors when deriving ``truly 2D'' numerical fluxes from local Green's function. A main ingredient for proving that such a property holds is the optimality of the trapezoidal rule for periodic functions. In accordance with practical evidence, a ``Bessel scheme'' previously introduced in [SIAM J. Numer. Anal. 56 (2018), pp. 2845-2870] is shown to be ``2D well-balanced'' (along with former algorithms known as ``discrete weighted means'' or ``tailored schemes''. Some $ L^2$ stability estimates are established, too.

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Laurent Gosse
Affiliation: IAC–CNR “Mauro Picone” (sezione di Roma) - Via dei Taurini, 19 - 00185 Rome, Italy

Received by editor(s): March 8, 2018
Received by editor(s) in revised form: November 18, 2018
Published electronically: July 16, 2019
Additional Notes: The support of Italian Minister of Instruction, University and Research (MIUR) through PRIN Project 2017, entitled “Innovative numerical methods for evolutionary partial differential equations and applications” #2017KKJP4X is acknowledged.
Article copyright: © Copyright 2019 American Mathematical Society