Anti‐diffusive alternate‐directions schemes for the transport of step functions

• Published in: International journal for numerical methods in fluids Vol. 94; no. 8; pp. 1155 - 1182
• Main Authors: Batteux, Léa, Duval, Fabien, Herbin, Raphaële, Latché, Jean‐Claude, Poullet, Pascal
• Format: Journal Article
• Language: English
• Published: Bognor Regis Wiley Subscription Services, Inc 01.08.2022; Wiley
• Subjects: Algorithms; anti‐diffusion; Diffusion; Dimensions; finite‐volume schemes; Mathematics; Maximum principle; Numerical Analysis; Plateaus; Step functions; Structured grids (mathematics); Transport; Transport equations; Velocity; 2010 AMS Subject Classification. -Please; give AMS classification codes - Finite-volume schemes; anti-diffusive scheme; transport equation; maximum principle
• ISSN: 0271-2091; 1097-0363
• DOI: 10.1002/fld.5086

The purpose in this article is to design finite‐volume schemes on structured grids for the transport of piecewise‐constant functions (typically, indicator functions) with as low diffusion as possible. We first propose an extension of the so‐called Lagrange‐projection algorithm, or downwind scheme with an Ultrabee limiter, for the transport equation in one space dimension with a non‐constant velocity; as its constant velocity counterpart, this scheme is designed to capture the discontinuities separating two plateaus in only one cell, and is referred to as “anti‐diffusive.” It is shown to preserve the bounds of the solution. Then, for two and three dimensional problems, we introduce a conservative alternate‐directions algorithm, an show that this latter enjoys a discrete maximum principle, provided that the underlying one‐dimensional schemes satisfy a property which may be seen as a flux limitation, possibly incorporated a posteriori in any explicit scheme. Numerical tests of this alternate‐directions algorithm are performed, with a variety of one‐dimensional embedded schemes including the anti‐diffusive scheme developed here and the so called THINC scheme. The observed numerical diffusion is indeed very low. With the anti‐diffusive scheme, the above‐mentioned a posteriori limitation is necessary to preserve the solution bounds, but, in the performed tests, does not introduce any visible additional diffusion. The purpose in this article is to design finite‐volumes schemes on structured grids for the transport of piecewise‐constant functions (typically, indicator functions) with as low diffusion as possible. We first propose an extension of the so‐called Lagrange‐projection algorithm for the transport equation in one space dimension with a non‐constant velocity which captures the discontinuities separating two plateaus in only one cell and preserves the bounds of the solution. Then, for two and three dimensional problems, we show how to embed this scheme (or any another one‐dimensional anti‐diffusive scheme) in an alternate‐directions algorithm while preserving both the conservativity and the bounds of the solution.