Close Search Advanced
Journals
Resources
About Us
Open Access

A Parameter-Free Generalized Moment Limiter for High-Order Methods on Unstructured Grids

A Parameter-Free Generalized Moment Limiter for High-Order Methods on Unstructured Grids
Preview
Add to basket

Year:    2009

Author:    Michael Yang, Z.J. Wang

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 4 : pp. 451–480

Abstract

A parameter-free limiting technique is developed for high-order unstruc- tured-grid methods to capture discontinuities when solving hyperbolic conservation laws. The technique is based on a "troubled-cell" approach, in which cells requiring limiting are first marked, and then a limiter is applied to these marked cells. A parameter-free accuracy-preserving TVD marker based on the cell-averaged solutions and solution derivatives in a local stencil is compared to several other markers in the literature in identifying "troubled cells". This marker is shown to be reliable and efficient to consistently mark the discontinuities. Then a compact high-order hierarchical moment limiter is developed for arbitrary unstructured grids. The limiter preserves a degree $p$ polynomial on an arbitrary mesh. As a result, the solution accuracy near smooth local extrema is preserved. Numerical results for the high-order spectral difference methods are provided to illustrate the accuracy, effectiveness, and robustness of the present limiting technique.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.09-m0913

Advances in Applied Mathematics and Mechanics, Vol. 1 (2009), Iss. 4 : pp. 451–480

Published online:    2009-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    30

Keywords:    Limiter shock-capturing high-order unstructured grids.

Author Details

Michael Yang

Z.J. Wang

  1. High-accurate and robust conservative remapping combining polynomial and hyperbolic tangent reconstructions

    Kucharik, Milan | Loubère, Raphaël

    Computers & Fluids, Vol. 208 (2020), Iss. P.104614

    https://doi.org/10.1016/j.compfluid.2020.104614 [Citations: 1]

  2. Analysis of slope limiters on unstructured triangular meshes

    Giuliani, Andrew | Krivodonova, Lilia

    Journal of Computational Physics, Vol. 374 (2018), Iss. P.1

    https://doi.org/10.1016/j.jcp.2018.07.031 [Citations: 13]

  3. Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics

    Vilar, François

    Computers & Fluids, Vol. 64 (2012), Iss. P.64

    https://doi.org/10.1016/j.compfluid.2012.05.001 [Citations: 54]

  4. Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics

    Vilar, François | Maire, Pierre-Henri | Abgrall, Rémi

    Computers & Fluids, Vol. 46 (2011), Iss. 1 P.498

    https://doi.org/10.1016/j.compfluid.2010.07.018 [Citations: 38]

  5. A parameter-free smoothness indicator for high-resolution finite element schemes

    Kuzmin, Dmitri | Schieweck, Friedhelm

    Open Mathematics, Vol. 11 (2013), Iss. 8

    https://doi.org/10.2478/s11533-013-0254-4 [Citations: 4]

  6. Discontinuous Galerkin formulation for 2D hydrodynamic modelling: Trade-offs between theoretical complexity and practical convenience

    Kesserwani, Georges | Ayog, Janice Lynn | Bau, Domenico

    Computer Methods in Applied Mechanics and Engineering, Vol. 342 (2018), Iss. P.710

    https://doi.org/10.1016/j.cma.2018.08.003 [Citations: 13]

  7. Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements

    Lohmann, Christoph | Kuzmin, Dmitri | Shadid, John N. | Mabuza, Sibusiso

    Journal of Computational Physics, Vol. 344 (2017), Iss. P.151

    https://doi.org/10.1016/j.jcp.2017.04.059 [Citations: 43]

  8. Unsteady Aerodynamic Applications Using High-Order Unstructured Grid Methods

    Breviglieri, Carlos | Azevedo, Joao Luiz

    50th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, (2012),

    https://doi.org/10.2514/6.2012-701 [Citations: 2]

  9. High-Order Solution-Adaptive Central Essentially Non-Oscillatory (CENO) Method for Viscous Flows

    Ivan, Lucian | Groth, Clinton

    49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, (2011),

    https://doi.org/10.2514/6.2011-367 [Citations: 10]

  10. Reduced one‐dimensional modelling and numerical simulation for mass transport in fluids

    Köppl, T. | Wohlmuth, B. | Helmig, R.

    International Journal for Numerical Methods in Fluids, Vol. 72 (2013), Iss. 2 P.135

    https://doi.org/10.1002/fld.3728 [Citations: 17]

  11. Hybrid Spectral Difference/Embedded MPWENO Method for Conservation Laws

    Choi, Jung J.

    53rd AIAA Aerospace Sciences Meeting, (2015),

    https://doi.org/10.2514/6.2015-0296 [Citations: 0]

  12. Symmetry-preserving WENO-type reconstruction schemes in Lagrangian hydrodynamics

    Liu, Xiaodong | Morgan, Nathaniel R.

    Computers & Fluids, Vol. 205 (2020), Iss. P.104528

    https://doi.org/10.1016/j.compfluid.2020.104528 [Citations: 7]

  13. FESTUNG: A MATLAB/GNU Octave toolbox for the discontinuous Galerkin method, Part II: Advection operator and slope limiting

    Reuter, Balthasar | Aizinger, Vadym | Wieland, Manuel | Frank, Florian | Knabner, Peter

    Computers & Mathematics with Applications, Vol. 72 (2016), Iss. 7 P.1896

    https://doi.org/10.1016/j.camwa.2016.08.006 [Citations: 18]
  14. Efficient Implementation of High-Order Accurate Numerical Methods on Unstructured Grids

    Accuracy Preserving Limiters for High-Order Finite Volume Methods

    Li, Wanai

    2014

    https://doi.org/10.1007/978-3-662-43432-1_3 [Citations: 0]

  15. Implicit High-Order Spectral Finite Volume Method for Inviscid Compressible Flows

    Breviglieri, Carlos | Azevedo, Joao Luiz F. | Basso, Edson | Souza, Maximiliano A. F.

    AIAA Journal, Vol. 48 (2010), Iss. 10 P.2365

    https://doi.org/10.2514/1.J050395 [Citations: 7]

  16. Implicit Spectral Difference Method Solutions of Compressible Flows Considering High-Order Meshes

    Moreira, Fabio M. | Jourdan, Eduardo | Breviglieri, Carlos | Aguiar, Andre R. | Azevedo, Joao Luiz F.

    46th AIAA Fluid Dynamics Conference, (2016),

    https://doi.org/10.2514/6.2016-3352 [Citations: 4]

  17. Space–time adaptive ADER discontinuous Galerkin finite element schemes with a posteriori sub-cell finite volume limiting

    Zanotti, Olindo | Fambri, Francesco | Dumbser, Michael | Hidalgo, Arturo

    Computers & Fluids, Vol. 118 (2015), Iss. P.204

    https://doi.org/10.1016/j.compfluid.2015.06.020 [Citations: 118]

  18. Anisotropic slope limiting for discontinuous Galerkin methods

    Aizinger, Vadym | Kosík, Adam | Kuzmin, Dmitri | Reuter, Balthasar

    International Journal for Numerical Methods in Fluids, Vol. 84 (2017), Iss. 9 P.543

    https://doi.org/10.1002/fld.4360 [Citations: 13]

  19. Hybrid spectral difference/embedded finite volume method for conservation laws

    Choi, Jung J.

    Journal of Computational Physics, Vol. 295 (2015), Iss. P.285

    https://doi.org/10.1016/j.jcp.2015.04.013 [Citations: 5]

  20. Localized Artificial Viscosity Stabilization of Discontinuous Galerkin Methods for Nonhydrostatic Mesoscale Atmospheric Modeling

    Yu, M. L. | Giraldo, F. X. | Peng, M. | Wang, Z. J.

    Monthly Weather Review, Vol. 143 (2015), Iss. 12 P.4823

    https://doi.org/10.1175/MWR-D-15-0134.1 [Citations: 24]

  21. Application of discontinuous Galerkin method in supersonic and hypersonic gas flows

    Wang, Donghuan | Lian, Yeda | Xiao, Hong

    Computers & Mathematics with Applications, Vol. 80 (2020), Iss. 1 P.227

    https://doi.org/10.1016/j.camwa.2020.03.013 [Citations: 2]

  22. A Comparative Study of Discontinuous High Order Methods for Compressible Flows

    Breviglieri, Carlos | Paula, Luis Gustavo L. | Wolf, William R. | Moreira, Fabio M. | Azevedo, Joao Luiz F.

    32nd AIAA Applied Aerodynamics Conference, (2014),

    https://doi.org/10.2514/6.2014-2984 [Citations: 2]

  23. Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration

    Vilar, François

    Journal of Computational Physics, Vol. 521 (2025), Iss. P.113535

    https://doi.org/10.1016/j.jcp.2024.113535 [Citations: 0]

  24. A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws

    Dumbser, Michael | Zanotti, Olindo | Loubère, Raphaël | Diot, Steven

    Journal of Computational Physics, Vol. 278 (2014), Iss. P.47

    https://doi.org/10.1016/j.jcp.2014.08.009 [Citations: 258]

  25. High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows

    Ivan, Lucian | Groth, Clinton P.T.

    Journal of Computational Physics, Vol. 257 (2014), Iss. P.830

    https://doi.org/10.1016/j.jcp.2013.09.045 [Citations: 69]

  26. The multi-dimensional limiters for solving hyperbolic conservation laws on unstructured grids

    Li, Wanai | Ren, Yu-Xin | Lei, Guodong | Luo, Hong

    Journal of Computational Physics, Vol. 230 (2011), Iss. 21 P.7775

    https://doi.org/10.1016/j.jcp.2011.06.018 [Citations: 49]

  27. High-order multi-dimensional limiting strategy with subcell resolution I. Two-dimensional mixed meshes

    You, Hojun | Kim, Chongam

    Journal of Computational Physics, Vol. 375 (2018), Iss. P.1005

    https://doi.org/10.1016/j.jcp.2018.09.011 [Citations: 29]

  28. Mathematical modeling of the multicomponent flow in porous media using higher-order methods

    Gális, Petr | Mikyška, Jiří

    Journal of Computational Physics, Vol. 493 (2023), Iss. P.112468

    https://doi.org/10.1016/j.jcp.2023.112468 [Citations: 0]

  29. Influence of Total-Variation-Diminishing Slope Limiting on Local Discontinuous Galerkin Solutions of the Shallow Water Equations

    Kesserwani, Georges | Liang, Qiuhua

    Journal of Hydraulic Engineering, Vol. 138 (2012), Iss. 2 P.216

    https://doi.org/10.1061/(ASCE)HY.1943-7900.0000494 [Citations: 9]

  30. High-Order Compressible Flow Simulations with the Unstructured Spectral Difference Method

    Moreira, Fabio M. | Breviglieri, Carlos | Azevedo, Joao Luiz F.

    22nd AIAA Computational Fluid Dynamics Conference, (2015),

    https://doi.org/10.2514/6.2015-3195 [Citations: 2]
  31. Handbook of Numerical Methods for Hyperbolic Problems - Basic and Fundamental Issues

    Spectral Volume and Spectral Difference Methods

    Wang, Z.J. | Liu, Y. | Lacor, C. | Azevedo, J.L.F.

    2016

    https://doi.org/10.1016/bs.hna.2016.09.013 [Citations: 2]

  32. Discontinuous Galerkin method with the spectral deferred correction time-integration scheme and a modified moment limiter for adaptive grids

    Gryngarten, Leandro | Smith, Andrew | Menon, Suresh

    Communications in Applied Mathematics and Computational Science, Vol. 7 (2012), Iss. 2 P.133

    https://doi.org/10.2140/camcos.2012.7.133 [Citations: 2]

  33. High-order computational fluid dynamics tools for aircraft design

    Wang, Z. J.

    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 372 (2014), Iss. 2022 P.20130318

    https://doi.org/10.1098/rsta.2013.0318 [Citations: 34]

  34. A Modified A Posteriori Subcell Limiter for High Order Flux Reconstruction Scheme for One-Dimensional Detonation Simulation

    Liu, Shiwei | Yuan, Li

    Journal of Scientific Computing, Vol. 97 (2023), Iss. 2

    https://doi.org/10.1007/s10915-023-02347-7 [Citations: 1]

  35. A high-order Lagrangian discontinuous Galerkin hydrodynamic method for quadratic cells using a subcell mesh stabilization scheme

    Liu, Xiaodong | Morgan, Nathaniel R. | Burton, Donald E.

    Journal of Computational Physics, Vol. 386 (2019), Iss. P.110

    https://doi.org/10.1016/j.jcp.2019.02.008 [Citations: 37]

  36. A comparison between bottom-discontinuity numerical treatments in the DG framework

    Caleffi, Valerio | Valiani, Alessandro | Li, Gang

    Applied Mathematical Modelling, Vol. 40 (2016), Iss. 17-18 P.7516

    https://doi.org/10.1016/j.apm.2015.09.025 [Citations: 14]

  37. A Posteriori Local Subcell Correction of High-Order Discontinuous Galerkin Scheme for Conservation Laws on Two-Dimensional Unstructured Grids

    Vilar, François | Abgrall, Rémi

    SIAM Journal on Scientific Computing, Vol. 46 (2024), Iss. 2 P.A851

    https://doi.org/10.1137/22M1542696 [Citations: 4]

  38. A Study of Efficiency of Implicit High-Order Spectral Difference Method Implementations

    Jourdan, Eduardo | Moreira, Fabio M. | Breviglieri, Carlos | Azevedo, Joao Luiz F. | Wang, Zhi J.

    55th AIAA Aerospace Sciences Meeting, (2017),

    https://doi.org/10.2514/6.2017-0740 [Citations: 3]

  39. A high-order moment limiter for the discontinuous Galerkin method on triangular meshes

    Dutt, Krishna | Krivodonova, Lilia

    Journal of Computational Physics, Vol. 433 (2021), Iss. P.110188

    https://doi.org/10.1016/j.jcp.2021.110188 [Citations: 5]

  40. Implementation of spectral difference method on overset grids for compressible inviscid flows

    Yang, Liu | Yang, Aiming

    Computers & Fluids, Vol. 140 (2016), Iss. P.500

    https://doi.org/10.1016/j.compfluid.2016.09.025 [Citations: 5]

  41. A conservative high‐order discontinuous Galerkin method for the shallow water equations with arbitrary topography

    Kesserwani, Georges | Liang, Qiuhua

    International Journal for Numerical Methods in Engineering, Vol. 86 (2011), Iss. 1 P.47

    https://doi.org/10.1002/nme.3044 [Citations: 26]