article

An unconditionally stable fully conservative semi-Lagrangian method

Authors:

Michael Lentine,

Jón Tómas Grétarsson,

Ronald FedkiwAuthors Info & Claims

Journal of Computational Physics, Volume 230, Issue 8

Pages 2857 - 2879

https://doi.org/10.1016/j.jcp.2010.12.036

Published: 01 April 2011 Publication History

Abstract

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.

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Colera MCarpio JBermejo R(2022)A nearly-conservative, high-order, forward Lagrange–Galerkin method for the resolution of compressible flows on unstructured triangular meshesJournal of Computational Physics10.1016/j.jcp.2022.111471467:COnline publication date: 15-Oct-2022
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Barrett AFogelson AGriffith B(2022)A hybrid semi-Lagrangian cut cell method for advection-diffusion problems with Robin boundary conditions in moving domainsJournal of Computational Physics10.1016/j.jcp.2021.110805449:COnline publication date: 15-Jan-2022
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cover image Journal of Computational Physics

Journal of Computational Physics Volume 230, Issue 8

April, 2011

516 pages

ISSN:0021-9991

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Copyright © Elsevier Inc. © 2011.

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Academic Press Professional, Inc.

United States

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Published: 01 April 2011

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Cited By

Colera MCarpio JBermejo R(2022)A nearly-conservative, high-order, forward Lagrange–Galerkin method for the resolution of compressible flows on unstructured triangular meshesJournal of Computational Physics10.1016/j.jcp.2022.111471467:COnline publication date: 15-Oct-2022
https://dl.acm.org/doi/10.1016/j.jcp.2022.111471
Barrett AFogelson AGriffith B(2022)A hybrid semi-Lagrangian cut cell method for advection-diffusion problems with Robin boundary conditions in moving domainsJournal of Computational Physics10.1016/j.jcp.2021.110805449:COnline publication date: 15-Jan-2022
https://dl.acm.org/doi/10.1016/j.jcp.2021.110805
Boscheri WTavelli MPareschi L(2022)On the Construction of Conservative Semi-Lagrangian IMEX Advection Schemes for Multiscale Time Dependent PDEsJournal of Scientific Computing10.1007/s10915-022-01768-090:3Online publication date: 1-Mar-2022
https://dl.acm.org/doi/10.1007/s10915-022-01768-0
Zhang JShen XGuo HFu HHan H(2020)Characteristic splitting mixed finite element analysis of compressible wormhole propagationApplied Numerical Mathematics10.1016/j.apnum.2019.08.012147:C(66-87)Online publication date: 1-Jan-2020
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Gibou FHyde DFedkiw R(2019)Sharp interface approaches and deep learning techniques for multiphase flowsJournal of Computational Physics10.1016/j.jcp.2018.05.031380:C(442-463)Online publication date: 1-Mar-2019
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Jalba ASobiecki ATelea A(2016)An Unified Multiscale Framework for Planar, Surface, and Curve SkeletonizationIEEE Transactions on Pattern Analysis and Machine Intelligence10.1109/TPAMI.2015.241442038:1(30-45)Online publication date: 1-Jan-2016
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Qiu LLu WFedkiw R(2016)An adaptive discretization of compressible flow using a multitude of moving Cartesian gridsJournal of Computational Physics10.1016/j.jcp.2015.10.025305:C(75-110)Online publication date: 15-Jan-2016
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He XWang HZhang FWang HWang GZhou KWu EBarbič JDeng Z(2015)Simulation of fluid mixing with interface controlProceedings of the 14th ACM SIGGRAPH / Eurographics Symposium on Computer Animation10.1145/2786784.2786791(129-135)Online publication date: 7-Aug-2015
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