An unconditionally stable fully conservative semi-Lagrangian method

Michael Lentine, Jón Tómas Grétarsson, Ronald Fedkiw

Research output: Contribution to journalArticlepeer-review

81 Scopus citations


Semi-Lagrangian methods have been around for some time, dating back at least to [3]. Researchers have worked to increase their accuracy, and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the CFL-based time step restriction of the smallest cell can be overwhelming. Since these schemes are based on characteristic tracing and interpolation, they do not readily lend themselves to a fully conservative implementation. However, we propose a novel technique that applies a conservative limiter to the typical semi-Lagrangian interpolation step in order to guarantee that the amount of the conservative quantity does not increase during this advection. In addition, we propose a new second step that forward advects any of the conserved quantity that was not accounted for in the typical semi-Lagrangian advection. We show that this new scheme can be used to conserve both mass and momentum for incompressible flows. For incompressible flows, we further explore properly conserving kinetic energy during the advection step, but note that the divergence free projection results in a velocity field which is inconsistent with conservation of kinetic energy (even for inviscid flows where it should be conserved). For compressible flows, we rely on a recently proposed splitting technique that eliminates the acoustic CFL time step restriction via an incompressible-style pressure solve. Then our new method can be applied to conservatively advect mass, momentum and total energy in order to exactly conserve these quantities, and remove the remaining time step restriction based on fluid velocity that the original scheme still had. © 2011 Elsevier Inc.
Original languageEnglish (US)
Pages (from-to)2857-2879
Number of pages23
JournalJournal of Computational Physics
Issue number8
StatePublished - Apr 2011
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): 42959
Acknowledgements: Research supported in part by ONR N0014-06-1-0393, ONR N00014-06-1-0505, ONR N00014-09-1-0101, ONR N00014-11-1-0027, ONR N00014-05-1-0479 for a computing cluster, and King Abdullah University of Science and Technology (KAUST) 42959. J.G. was supported in part by, and computational resources were provided in part by ONR N00014-06-1-0505 and ONR N00014-09-C-015. M.L. was supported in part by an Intel Ph.D. Fellowship.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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