A single-stage flux-corrected transport algorithm for high-order finite-volume methods

Chaplin, Christopher; Colella, Phillip

doi:10.2140/camcos.2017.12.1

Title: A single-stage flux-corrected transport algorithm for high-order finite-volume methods

Journal Article · Mon May 08 00:00:00 EDT 2017 · Communications in Applied Mathematics and Computational Science

DOI:https://doi.org/10.2140/camcos.2017.12.1· OSTI ID:1420104

Chaplin, Christopher ^[1]; Colella, Phillip ^[1]

Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Applied Numerical Algorithms Group, Computational Research Division

We present a new limiter method for solving the advection equation using a high-order, finite-volume discretization. The limiter is based on the flux-corrected transport algorithm. Here, we modify the classical algorithm by introducing a new computation for solution bounds at smooth extrema, as well as improving the preconstraint on the high-order fluxes. We compute the high-order fluxes via a method-of-lines approach with fourth-order Runge-Kutta as the time integrator. For computing low-order fluxes, we select the corner-transport upwind method due to its improved stability over donor-cell upwind. Several spatial differencing schemes are investigated for the high-order flux computation, including centered- difference and upwind schemes. We show that the upwind schemes perform well on account of the dissipation of high-wavenumber components. The new limiter method retains high-order accuracy for smooth solutions and accurately captures fronts in discontinuous solutions. Further, we need only apply the limiter once per complete time step.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: AC02-05CH11231

OSTI ID:: 1420104

Journal Information:: Communications in Applied Mathematics and Computational Science, Vol. 12, Issue 1; ISSN 1559-3940

Publisher:: Mathematical Sciences PublishersCopyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 3 works

Citation information provided by
Web of Science

References (26)

Uniformly high order accurate essentially non-oscillatory schemes, III Harten, Ami; Engquist, Bjorn; Osher, Stanley Journal of Computational Physics, Vol. 71, Issue 2 https://doi.org/10.1016/0021-9991(87)90031-3	journal	August 1987
High-resolution FEM–FCT schemes for multidimensional conservation laws Kuzmin, D.; Möller, M.; Turek, S. Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 45-47 https://doi.org/10.1016/j.cma.2004.05.009	journal	November 2004
ADER schemes for three-dimensional non-linear hyperbolic systems Titarev, V. A.; Toro, E. F. Journal of Computational Physics, Vol. 204, Issue 2 https://doi.org/10.1016/j.jcp.2004.10.028	journal	April 2005
ADER-WENO finite volume schemes with space–time adaptive mesh refinement Dumbser, Michael; Zanotti, Olindo; Hidalgo, Arturo Journal of Computational Physics, Vol. 248 https://doi.org/10.1016/j.jcp.2013.04.017	journal	September 2013
High resolution schemes for hyperbolic conservation laws Harten, Ami Journal of Computational Physics, Vol. 49, Issue 3 https://doi.org/10.1016/0021-9991(83)90136-5	journal	March 1983
A limiter for PPM that preserves accuracy at smooth extrema Colella, Phillip; Sekora, Michael D. Journal of Computational Physics, Vol. 227, Issue 15 https://doi.org/10.1016/j.jcp.2008.03.034	journal	July 2008
One-dimensional shock-capturing for high-order discontinuous Galerkin methods: ONE-DIMENSIONAL SHOCK-CAPTURING FOR HIGH-ORDER DG METHODS Casoni, E.; Peraire, J.; Huerta, A. International Journal for Numerical Methods in Fluids, Vol. 71, Issue 6 https://doi.org/10.1002/fld.3682	journal	October 2012
High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems Shu, Chi-Wang SIAM Review, Vol. 51, Issue 1 https://doi.org/10.1137/070679065	journal	February 2009
Finite element flux-corrected transport (FEM-FCT) for the euler and Navier-Stokes equations Löhner, Rainald; Morgan, Ken; Peraire, Jaime International Journal for Numerical Methods in Fluids, Vol. 7, Issue 10 https://doi.org/10.1002/fld.1650071007	journal	October 1987
Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method van Leer, Bram Journal of Computational Physics, Vol. 32, Issue 1 https://doi.org/10.1016/0021-9991(79)90145-1	journal	July 1979
Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes – Speed comparisons with Runge–Kutta methods Balsara, Dinshaw S.; Meyer, Chad; Dumbser, Michael Journal of Computational Physics, Vol. 235 https://doi.org/10.1016/j.jcp.2012.04.051	journal	February 2013
Hierarchical slope limiting in explicit and implicit discontinuous Galerkin methods Kuzmin, Dmitri Journal of Computational Physics, Vol. 257 https://doi.org/10.1016/j.jcp.2013.04.032	journal	January 2014
An Unsplit 3D Upwind Method for Hyperbolic Conservation Laws Saltzman, Jeff Journal of Computational Physics, Vol. 115, Issue 1 https://doi.org/10.1006/jcph.1994.1184	journal	November 1994
Weighted Essentially Non-oscillatory Schemes Liu, Xu-Dong; Osher, Stanley; Chan, Tony Journal of Computational Physics, Vol. 115, Issue 1 https://doi.org/10.1006/jcph.1994.1187	journal	November 1994
Fully multidimensional flux-corrected transport algorithms for fluids Zalesak, Steven T. Journal of Computational Physics, Vol. 31, Issue 3 https://doi.org/10.1016/0021-9991(79)90051-2	journal	June 1979
A high-order finite-volume method for conservation laws on locally refined grids McCorquodale, Peter; Colella, Phillip Communications in Applied Mathematics and Computational Science, Vol. 6, Issue 1 https://doi.org/10.2140/camcos.2011.6.1	journal	January 2011
The Piecewise Parabolic Method (PPM) for gas-dynamical simulations Colella, Phillip; Woodward, Paul R. Journal of Computational Physics, Vol. 54, Issue 1 https://doi.org/10.1016/0021-9991(84)90143-8	journal	April 1984
FEM-FCT: Combining unstructured grids with high resolution Löhner, R.; Morgan, K.; Vahdati, M. Communications in Applied Numerical Methods, Vol. 4, Issue 6 https://doi.org/10.1002/cnm.1630040605	journal	November 1988
Multidimensional upwind methods for hyperbolic conservation laws Colella, Phillip Journal of Computational Physics, Vol. 87, Issue 1 https://doi.org/10.1016/0021-9991(90)90233-Q	journal	March 1990
Solution of the generalized Riemann problem for advection–reaction equations Toro, E. F.; Titarev, V. A. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, Vol. 458, Issue 2018 https://doi.org/10.1098/rspa.2001.0926	journal	January 2002
A vertex-based hierarchical slope limiter for <mml:math altimg="si27.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd"><mml:mi>p</mml:mi></mml:math>-adaptive discontinuous Galerkin methods Kuzmin, Dmitri Journal of Computational and Applied Mathematics, Vol. 233, Issue 12 https://doi.org/10.1016/j.cam.2009.05.028	journal	April 2010
Anti-diffusive flux corrections for high order finite difference WENO schemes Xu, Zhengfu; Shu, Chi-Wang Journal of Computational Physics, Vol. 205, Issue 2 https://doi.org/10.1016/j.jcp.2004.11.014	journal	May 2005
High-order, finite-volume methods in mapped coordinates Colella, P.; Dorr, M. R.; Hittinger, J. A. F. Journal of Computational Physics, Vol. 230, Issue 8 https://doi.org/10.1016/j.jcp.2010.12.044	journal	April 2011
Roofline: an insightful visual performance model for multicore architectures Williams, Samuel; Waterman, Andrew; Patterson, David Communications of the ACM, Vol. 52, Issue 4 https://doi.org/10.1145/1498765.1498785	journal	April 2009
Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works Boris, Jay P.; Book, David L. Journal of Computational Physics, Vol. 11, Issue 1 https://doi.org/10.1016/0021-9991(73)90147-2	journal	January 1973
A Front Tracking Method for Compressible Flames in One Dimension Hilditch, James; Colella, Phillip SIAM Journal on Scientific Computing, Vol. 16, Issue 4 https://doi.org/10.1137/0916045	journal	July 1995

Similar Records

A positive finite-difference advection scheme

Journal Article · Wed Mar 01 00:00:00 EST 1995 · Journal of Computational Physics · OSTI ID:1420104

Hundsdorfer, W; Koren, B; Loon, M van

A high-performance finite-volume algorithm for solving partial differential equations governing compressible viscous flows on structured grids

Journal Article · Tue Nov 01 00:00:00 EDT 2016 · Computers and Mathematics with Applications (Oxford) · OSTI ID:1420104

Guzik, S. M.; Gao, X.; Olschanowsky, C.

Matrix-free subcell residual distribution for Bernstein finite element discretizations of linear advection equations

Journal Article · Wed Oct 09 00:00:00 EDT 2019 · Computer Methods in Applied Mechanics and Engineering · OSTI ID:1420104

Hajduk, Hennes; Kuzmin, Dmitri; Kolev, Tzanio; +1 more

Related Subjects

97 MATHEMATICS AND COMPUTING
finite-volume method
high order
advection
limiter

Title: A single-stage flux-corrected transport algorithm for high-order finite-volume methods

Citation Formats

References (26)

Similar Records

Related Subjects