Hyperbolic Method

Hyperbolize, Discretize, and Solve.





▶ Home

▶ History
▶ Development
▶ FAQ
▶ Related Topics
Development



Finite-Volume (FV):


It aims at improving current practical CFD codes, many of which are based on FV methods. The focus here is on the node-centered edge-based FV scheme, which has a special property on triangles/tetrahedra of achieving third-order accuracy. It results in a third-order NS solver that converges intrinsically faster than conventional second-order NS solver widely used in the current CFD codes.

  1. 3rd-order low-dissipation scheme for practical CFD codes

    Hiroaki Nishikawa and Yi Liu, "Third-Order Edge-Based Scheme for Unsteady Problems", AIAA Paper 2018-4166, AIAA 2018 Fluid Dynamics Conference, 25 - 29 June 2018, Atlanta, Georgia.
    [ bib | pdf | slides ]
  2. HNS20G: New Hyperbolic Navier-Stokes Formulation for Higher-Order

    Lingquan Li and Jialin Lou and Hong Luo and Hiroaki Nishikawa, "A New Formulation of Hyperbolic Navier-Stokes Solver based on Finite Volume Method on Arbitrary Grids", AIAA Paper 2018-4160, AIAA 2018 Fluid Dynamics Conference, 25 - 29 June 2018, Atlanta, Georgia.
    [ bib | pdf ]
  3. Hyperbolic Scheme is Exact for Piecewise-Linear Solution

    H. Nishikawa, "On Hyperbolic Method for Diffusion with Discontinuous Coefficients", Journal of Computational Physics, Volume 367, 2018, Pages 102-108, 2018.
    [ bib | pdf | journal ]
  4. Proper Scaling for Length Scale

    H. Nishikawa and Y. Nakashima, "Dimensional Scaling and Numerical Similarity in Hyperbolic Method for Diffusion", Journal of Computational Physics, Volume 355, January 2018, Pages 121-143, 2018.
    [ bib | pdf | journal ]
  5. Analysis on High-Reynolds-Number Boundary layer

    H. Nishikawa and Y. Liu, "Hyperbolic Advection-Diffusion Schemes for High-Reynolds-Number Boundary-Layer Problems", Journal of Computational Physics, Volume 352, January 2018, Pages 23-51, 2018.
    [ bib | pdf | journal ] --- See also AIAA2017-0081 [ pdf | slides ]
  6. Third-Order Accuracy with Zero/Negative-Volume Elements

    H. Nishikawa, "Uses of Zero and Negative Volume Elements for Node-Centered Edge-Based Discretization", AIAA Paper 2017-4295, 23rd AIAA Computational Fluid Dynamics Conference, 5 - 9 June 2017, Denver, Colorado.
    [ bib | pdf | slides ]
  7. Third-order hyperbolic Navier-Stokes solver in FUN3D.

    Y. Liu and H. Nishikawa, "Third-Order Edge-Based Hyperbolic Navier-Stokes Scheme for Three-Dimensional Viscous Flows", AIAA Paper 2017-3443, 23rd AIAA Computational Fluid Dynamics Conference, 5 - 9 June 2017, Denver, Colorado.
    [ bib | pdf | slides ]
  8. NASA's 3D hyperbolic Navier-Stokes solver (Unsteady).

    Y. Liu and H. Nishikawa, "Third-Order Inviscid and Second-Order Hyperbolic Navier-Stokes Solvers for Three-Dimensional Unsteady Inviscid and Viscous Flows", AIAA Paper 2017-0738, 55th AIAA Aerospace Sciences Meeting, 9 - 13 January 2017, Grapevine, Texas.
    [ bib | pdf | slides ]
  9. NASA's 3D hyperbolic Navier-Stokes solver (Steady).

    Y. Liu and H. Nishikawa, "Third-Order Inviscid and Second-Order Hyperbolic Navier-Stokes Solvers for Three-Dimensional Inviscid and Viscous Flows", AIAA Paper 2016-3969, 46th AIAA Fluid Dynamics Conference, 13-17 June 2016, Washington, D.C.
    [ bib | pdf | slides ]
  10. 3D hyperbolic Navier-Stokes solver.

    Y. Nakashima, N. Watanabe, H. Nishikawa, "Hyperbolic Navier-Stokes Solver for Three-Dimensional Flows", AIAA Paper 2016-1101, 54th AIAA Aerospace Sciences Meeting, 4-8 January, San Diego, California, 2016.
    [ bib | pdf ]
  11. A practical hyperbolic Navier-Stokes system (HNS20).

    H. Nishikawa, "Alternative Formulations for First-, Second-, and Third-Order Hyperbolic Navier-Stokes Schemes", AIAA Paper 2015-2451, 22nd AIAA Computational Fluid Dynamics Conference, Dallas, 2015.
    [ bib | pdf | slides | seminar video ]
  12. Third-order accuracy without high-order curved grids.

    H. Nishikawa, " Accuracy-preserving boundary flux quadrature for finite-volume discretization on unstructured grids", Journal of Computational Physics, Volume 281, January 2015, Pages 518-555, 2015.
    [ bib | pdf | journal | slides | seminar video ]
  13. Third-order compressible/incompressible NS, Fully implicit. Extension to incompressible Navier-Stokes.

    H. Nishikawa, "First, Second, and Third Order Finite-Volume Schemes for Navier-Stokes Equations", AIAA Paper 2014-2091, 7th AIAA Theoretical Fluid Mechanics Conference, Atlanta, 2014.
    [ bib | pdf | seminar video ]
  14. Third-order advection-diffusion, Fully implicit.

    H. Nishikawa, First, Second, and Third Order Finite-Volume Schemes for Advection-Diffusion, Journal of Computational Physics, Volume 273, September 2014, Pages 287-309, 2014.
    [ bib | pdf | journal | seminar video ]
  15. Energy-stable 1st-order diffusion scheme, 3rd-order gradients.

    H. Nishikawa, First, Second, and Third Order Finite-Volume Schemes for Diffusion, Journal of Computational Physics, Volume 256, Issue 1, January 2014, Pages 791-805, 2014.
    [ bib | pdf | journal | seminar video | a derivation of Lr (pdf) ]
  16. Reformulate source term to achieve third-order accuracy easily.

    H. Nishikawa, Divergence Formulation of Source Term, Journal of Computational Physics, Volume 231, Issue 19, 1 August 2012, Pages 6393-6400, 2012.
    [ bib | pdf | journal | seminar video ]
  17. Extension to compressible Navier-Stokes, 2nd-order explicit.

    H. Nishikawa, New-Generation Hyperbolic Navier-Stokes Schemes:
    O(1/h) Speed-Up and Accurate Viscous/Heat Fluxes,
    AIAA Paper 2011-3043, 20th AIAA Computational Fluid Dynamics Conference, Hawaii, 2011.
    [ bib | pdf ]




Discontinuous Galerkin:


Hyperbolic method is applicable to Discontinuous Galerkin (DG) methods. The construction pursued for DG is unique in that the number of unknowns can be reduced despite the increased number of equations in the target PDEs by extedning Scheme-II .

  1. Surprisingloy, explicit hyperbolic diffusion schemes can be time accurate.

    Jialin Lou and Lingquan Li and Hong Luo and Hiroaki Nishikawa, "Explicit Hyperbolic Reconstructed Discontinuous {G}alerkin Methods for Time-Dependent Problems", AIAA Paper 2018-4270, AIAA 2018 Fluid Dynamics Conference, 25 - 29 June 2018, Atlanta, Georgia.
    [ bib | pdf ]
  2. Hyperbolic DG/rDG/FV for Unsteady Advection-Diffusion

    Jialin Lou, Lingquan Li, Hong Luo, and Hiroaki Nishikawa, "Reconstructed discontinuous Galerkin methods for linear advection-diffusion equations based on first-order hyperbolic system", Journal of Computational Physics, Volume 369, 2018, Pages 103-124, 2018.
    [ bib | pdf | journal ]
  3. New Hyperbolic Diffusion Formulation for High-Order DG/rDG.

    Jialin Lou, Lingquan Li, Hong Luo, Hiroaki Nishikawa, "First-Order Hyperbolic System Based Reconstructed Discontinuous Galerkin Methods for Nonlinear Diffusion Equations on Unstructured Grids", AIAA Paper 2018-2094, 56th AIAA Aerospace Sciences Meeting, 8 - 12 January 2018, Kissimmee, Florida.
    [ bib | pdf ]
  4. Hyperbolic DG and rDG for Advection Diffusion.

    Jialin Lou, Lingquan Li, Xiaodong Liu, Hong Luo, Hiroaki Nishikawa, "Reconstructed Discontinuous Galerkin Methods Based on First-Order Hyperbolic System for Advection-Diffusion Equations", AIAA Paper 2017-3445, 23rd AIAA Computational Fluid Dynamics Conference, 5 - 9 June 2017, Denver, Colorado.
    [ bib | pdf ]
  5. Hyperbolic DG/FV/rDG for Diffusion

    Jialin Lou and Xiaodong Liu and Hong Luo and Hiroaki Nishikawa, Reconstructed Discontinuous Galerkin Methods for Hyperbolic Diffusion Equations on Unstructured Grids, AIAA Paper 2017-0310, 55th AIAA Aerospace Sciences Meeting, 9 - 13 January 2017, Grapevine, Texas.
    [ bib | pdf ]
  6. Efficient High-Order Discontinuous Galerkin Schemes with First-Order Hyperbolic Advection-Diffusion System Approach,
    Journal of Computational Physics, Volume 321, 15 September 2016, Pages 729-754.
    [ bib | pdf | journal ]

    NOTE: The resulting hyperbolic DG scheme achieves the same order of accuracy in the advective term and the gradients (but one order lower for the diffusion term) as a standard DG scheme for the same number of degrees of freedom. See these personal notes, from which the work is originated, or JCP2018 [ pdf | journal ].
  7. Hyperbolic reconstructed-DG/FV/DG

    Jialin Lou, Hong Luo, and Hiroaki Nishikawa, Discontinuous Galerkin Methods for Hyperbolic Advection-Diffusion Equation on Unstructured Grids, The 9th International Conference on Computational Fluid Dynamics, July 11-15, Istanbul, Turkey, 2016.
    [ bib | pdf ]

    This is the origin of the development of high-order hyperbolic Navier-Stokes schemes based on reconstructed discontinuous Galekrin (rDG) methods, which is a general framework including finite-volume and discontinuous Galerkin methods as special cases.




Active-Flux:


Active-flux method is a compact high-order cell-centered FV scheme for unstructured grids. The cell-averaged solution is updated by the flux that evolves independently at cell faces. It achieves third-order accuracy within a compact stencil, with a much reduced number of degrees of freedom compared with DG methods. Upwinding mechanism can be naturally built in the computation of the fluxes, and thus suitable for hyperbolic systems. Consequently, it is suitable for the hyperbolic diffusion.

  1. Third-order time-accurate scheme for advection diffusion

    H. Nishikawa and P. L. Roe, Third-order active-flux scheme for advection diffusion: Hyperbolic diffusion, boundary condition, and Newton solver, Computers and Fluids, 125, pp.71-81 2016.
    [ bib | pdf | journal | slides | seminar video | slides2 ]




Residual-Distribution (RD):

RD method allows the construction of second-order schemes within a compact stencil. Hence, it is possible to develop Newton's method for solving the discrete problem, which converges to machine zero within at most 10 iterations. Successful development of a hyperbolic NS solver based on the RD method would enable extremely efficient CFD computations along with superior accuracy in the derivatives (e.g., the viscous stresses, heat fluxes, vorticity, etc).

  1. Hyperbolic RD scheme for Dispersion

    A First-Order Hyperbolic System Approach for Dispersion, Journal of Computational Physics, Volume 321, 15 September 2016, Pages 593-605.
    [ bib | pdf | journal ]

    AIAA Paper 2016-1331, 54th AIAA Aerospace Sciences Meeting, 4-8 January, San Diego, California, 2016.
    [pdf]
  2. Shock-capturing hyperbolic RD schemes

    High-order shock-capturing hyperbolic residual-distribution schemes on irregular triangular grids, Computers and Fluids, Volume 131, 5 June 2016, Pages 29-44, 2016.
    [ bib | journal ]

    High-Order Residual-Distribution Schemes for Discontinuous Problems on Irregular Triangular Grids, AIAA Paper 2016-1331, 54th AIAA Aerospace Sciences Meeting, 4-8 January, San Diego, California, 2016.
    [ bib | pdf ]
  3. A new design princicple for hyperbolic RD schemes

    Improved second-order hyperbolic residual-distribution scheme and its extension to third-order on arbitrary triangular grids, Journal of Computational Physics, 300, pp.455-491, 2015.
    [ bib | pdf | journal ]

    High-Order Hyperbolic Residual-Distribution Schemes on Arbitrary Triangular Grids, AIAA Paper 2015-2445, 22nd AIAA Computational Fluid Dynamics Conference, Dallas, 2015.
    [ bib | pdf ]
  4. High-order time-accurate schemes.

    Very efficient high-order hyperbolic schemes for time-dependent advection-diffusion problems: Third-, fourth-, and sixth-order, Computers and Fluids, 102, pp.131-147 2014.
    [ bib | pdf | journal | slides ]
  5. Extension to time-accurate scheme.

    First-Order Hyperbolic System Method for Time-Dependent Advection-Diffusion Problems, NASA-TM-2014-218175, March 2014.
    [ bib | pdf ]
  6. First extension to advection-diffusion, 2nd-order explicit.

    H. Nishikawa, A First-Order System Approach for Diffusion Equation. II: Unification of Advection and Diffusion, Journal of Computational Physics, 227, pp. 3989-4016, 2010.
    [ bib | pdf | journal ]
  7. First paper of the hyperbolic method, 2nd-order explicit.

    H. Nishikawa, A First-Order System Approach for Diffusion Equation. I: Second-Order Residual Distribution Schemes, Journal of Computational Physics, 227, pp. 315-352, 2007.
    [ bib | pdf | journal | code ]




Others:


Other works related to the hyperbolic method.
[ Please let us know if we're missing any work. ]

  1. Hyperbolic diffusion for distance function computations

    Rob Watson and Will Trojak and Paul G. Tucker, A Simple Flux Reconstruction Approach to Solving a Poisson Equation to find Wall Distances for Turbulence Modelling, AIAA Paper 2018-4166, AIAA 2018 Fluid Dynamics Conference, 25 - 29 June 2018, Atlanta, Georgia.
    [ AIAA ]
  2. Hyperbolic Navier-Stokes Finite-Volume Solver

    Tsukasa Nagao, Atsushi Hashimoto, and Tetsuya Sato, A Study on Time Evolution Method for Hyperbolic Navier-Stokes System, 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-0370) .
    [ AIAA ]
  3. Hyperbolic Method for Plasmas

    Rei Kawashima, Zhexu Wang, Amareshwara Sainadh Chamarthi, Hiroyuki Koizumi, Kimiya Komurasaki, Hyperbolic System Approach for Magnetized Electron Fluids in ExB Discharge Plasmas 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-0175) .
    [ AIAA ]
  4. High-Order Compact Schemes for Magnetizd Electron Fluid

    Amareshwara Sainadh Chamarthi and Zhexu Wang and Rei Kawashima, Weighted Nonlinear Schemes for Magnetized Electron Fluid in Quasi-neutral plasma, 2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 2018-2194) .
    [ AIAA ]
  5. Hyperbolic Navier-Stokes Fiite-Volume Solver for RANS

    Zhihao Wu and Dingxi Wang, The Development of a Hyperbolic {RANS} System for Analysing Turbomachinery Flow Field, Proc. of Shanghai 2017 Global Power and Propulsion Forum, GPPS-17-0198, 2017, Shanghai, China.
    [ pdf ]
  6. Hyperbolic method for MHD.

    Hubert Baty and Hiroaki Nishikawa, Hyperbolic method for magnetic reconnection process in steady state magnetohydrodynamics, Monthly Notices of the Royal Astronomical Society, Volume 459, June 11, 2016, Pages 624-637.
    [ Journal ]
  7. Hyperbolic method for anisotropic diffusion problems.

    Rei Kawashima, Kimiya Komurasaki, Tony Schönherra, A hyperbolic-equation system approach for magnetized electron fluids in quasi-neutral plasmas, Journal of Computational Physics, Volume 284, Issue 1, March 2015, Pages 59-69.
    [ Journal ]
  8. Entropy consitent Navier_Stokes methods via the hyperbolic method.

    Akmal Nizam Mohammed and Farzad Ismail, Entropy Consistent Methods for the Navier-Stokes Equations, Journal of Scientific Computing, August 2014, Pages 1-20.
    [ Journal ]





Copyright 2014- . Hiroaki Nishikawa. All rights reserved.