

Development
FiniteVolume (FV):
It aims at improving current practical CFD codes, many of which are based on
FV methods. The focus here is on the nodecentered edgebased FV scheme, which
has a special property on triangles/tetrahedra of achieving thirdorder accuracy.
It results in a thirdorder NS solver that converges intrinsically faster than conventional
secondorder NS solver widely used in the current CFD codes.

3rdorder lowdissipation scheme for practical CFD codes
Hiroaki Nishikawa and Yi Liu,
"ThirdOrder EdgeBased Scheme for Unsteady Problems",
AIAA Paper 20184166, AIAA 2018 Fluid Dynamics Conference, 25  29 June 2018, Atlanta, Georgia.
[ bib 
pdf 
slides
]

HNS20G: New Hyperbolic NavierStokes Formulation for HigherOrder
Lingquan Li and Jialin Lou and Hong Luo and Hiroaki Nishikawa,
"A New Formulation of Hyperbolic NavierStokes Solver based on Finite Volume Method on Arbitrary Grids",
AIAA Paper 20184160, AIAA 2018 Fluid Dynamics Conference, 25  29 June 2018, Atlanta, Georgia.
[ bib 
pdf
]

Hyperbolic Scheme is Exact for PiecewiseLinear Solution
H. Nishikawa, "On Hyperbolic Method for Diffusion with Discontinuous Coefficients",
Journal of Computational Physics, Volume 367, 2018, Pages 102108, 2018.
[ bib 
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]

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 121143, 2018.
[ bib 
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Analysis on HighReynoldsNumber Boundary layer
H. Nishikawa and Y. Liu, "Hyperbolic AdvectionDiffusion Schemes for HighReynoldsNumber BoundaryLayer Problems",
Journal of Computational Physics, Volume 352, January 2018, Pages 2351, 2018.
[ bib 
pdf  journal
]  See also AIAA20170081 [ pdf 
slides ]

ThirdOrder Accuracy with Zero/NegativeVolume Elements
H. Nishikawa, "Uses of Zero and Negative Volume Elements for NodeCentered EdgeBased Discretization",
AIAA Paper 20174295, 23rd AIAA Computational Fluid Dynamics Conference, 5  9 June 2017, Denver, Colorado.
[ bib 
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slides
]

Thirdorder hyperbolic NavierStokes solver in FUN3D.
Y. Liu and H. Nishikawa, "ThirdOrder EdgeBased Hyperbolic NavierStokes Scheme for ThreeDimensional Viscous Flows",
AIAA Paper 20173443, 23rd AIAA Computational Fluid Dynamics Conference, 5  9 June 2017, Denver, Colorado.
[ bib 
pdf 
slides
]

NASA's 3D hyperbolic NavierStokes solver (Unsteady).
Y. Liu and H. Nishikawa, "ThirdOrder Inviscid and SecondOrder Hyperbolic NavierStokes Solvers for ThreeDimensional Unsteady Inviscid and Viscous Flows",
AIAA Paper 20170738, 55th AIAA Aerospace Sciences Meeting, 9  13 January 2017, Grapevine, Texas.
[ bib 
pdf 
slides
]

NASA's 3D hyperbolic NavierStokes solver (Steady).
Y. Liu and H. Nishikawa, "ThirdOrder Inviscid and SecondOrder Hyperbolic NavierStokes Solvers
for ThreeDimensional Inviscid and Viscous Flows", AIAA Paper 20163969, 46th AIAA Fluid Dynamics Conference, 1317 June 2016, Washington, D.C.
[ bib 
pdf 
slides
]

3D hyperbolic NavierStokes solver.
Y. Nakashima, N. Watanabe, H. Nishikawa, "Hyperbolic NavierStokes Solver for ThreeDimensional Flows",
AIAA Paper 20161101, 54th AIAA Aerospace Sciences Meeting, 48 January, San Diego, California, 2016.
[ bib 
pdf
]

A practical hyperbolic NavierStokes system (HNS20).
H. Nishikawa, "Alternative Formulations for First, Second, and ThirdOrder Hyperbolic NavierStokes Schemes", AIAA Paper 20152451,
22nd AIAA Computational Fluid Dynamics Conference, Dallas, 2015.
[ bib 
pdf 
slides 
seminar video
]

Thirdorder accuracy without highorder curved grids.
H. Nishikawa, " Accuracypreserving boundary flux quadrature for finitevolume discretization on unstructured grids", Journal of Computational Physics,
Volume 281, January 2015, Pages 518555, 2015.
[ bib 
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slides 
seminar video
]

Thirdorder compressible/incompressible NS, Fully implicit.
Extension to incompressible NavierStokes.
H. Nishikawa, "First, Second, and Third Order FiniteVolume Schemes for NavierStokes Equations", AIAA Paper 20142091,
7th AIAA Theoretical Fluid Mechanics Conference, Atlanta, 2014.
[ bib 
pdf 
seminar video
]

Thirdorder advectiondiffusion, Fully implicit.
H. Nishikawa, First, Second, and Third Order FiniteVolume Schemes for AdvectionDiffusion, Journal of Computational Physics,
Volume 273, September 2014, Pages 287309, 2014.
[ bib 
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seminar video ]
 Energystable 1storder diffusion scheme, 3rdorder gradients.
H. Nishikawa, First, Second, and Third Order FiniteVolume Schemes for Diffusion, Journal of Computational Physics,
Volume 256, Issue 1, January 2014, Pages 791805, 2014.
[ bib  pdf 
journal 
seminar video 
a derivation of Lr (pdf) ]

Reformulate source term to achieve thirdorder accuracy easily.
H. Nishikawa, Divergence Formulation of Source Term, Journal of Computational Physics,
Volume 231, Issue 19, 1 August 2012, Pages 63936400, 2012.
[ bib  pdf 
journal 
seminar video
]

Extension to compressible NavierStokes, 2ndorder explicit.
H. Nishikawa, NewGeneration Hyperbolic NavierStokes Schemes:
O(1/h) SpeedUp and Accurate Viscous/Heat Fluxes, AIAA Paper 20113043,
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 SchemeII .

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 TimeDependent Problems",
AIAA Paper 20184270, AIAA 2018 Fluid Dynamics Conference, 25  29 June 2018, Atlanta, Georgia.
[ bib 
pdf
]

Hyperbolic DG/rDG/FV for Unsteady AdvectionDiffusion
Jialin Lou, Lingquan Li, Hong Luo, and Hiroaki Nishikawa,
"Reconstructed discontinuous Galerkin methods for linear advectiondiffusion equations based on firstorder hyperbolic system",
Journal of Computational Physics, Volume 369, 2018, Pages 103124, 2018.
[ bib 
pdf 
journal
]

New Hyperbolic Diffusion Formulation for HighOrder DG/rDG.
Jialin Lou, Lingquan Li, Hong Luo, Hiroaki Nishikawa, "FirstOrder Hyperbolic System Based Reconstructed Discontinuous Galerkin Methods for Nonlinear Diffusion Equations on Unstructured Grids",
AIAA Paper 20182094, 56th AIAA Aerospace Sciences Meeting, 8  12 January 2018, Kissimmee, Florida.
[ bib 
pdf
]

Hyperbolic DG and rDG for Advection Diffusion.
Jialin Lou, Lingquan Li, Xiaodong Liu, Hong Luo, Hiroaki Nishikawa,
"Reconstructed Discontinuous Galerkin Methods Based on FirstOrder Hyperbolic System for AdvectionDiffusion Equations",
AIAA Paper 20173445, 23rd AIAA Computational Fluid Dynamics Conference, 5  9 June 2017, Denver, Colorado.
[ bib 
pdf
]
 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 20170310, 55th AIAA Aerospace Sciences Meeting, 9  13 January 2017, Grapevine, Texas.
[ bib 
pdf
]

Efficient HighOrder Discontinuous Galerkin Schemes
with FirstOrder Hyperbolic AdvectionDiffusion System Approach,
Journal of Computational Physics, Volume 321, 15 September 2016, Pages 729754.
[ bib 
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]
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
].

Hyperbolic reconstructedDG/FV/DG
Jialin Lou, Hong Luo, and Hiroaki Nishikawa,
Discontinuous Galerkin Methods for Hyperbolic AdvectionDiffusion Equation on Unstructured Grids,
The 9th International Conference on Computational Fluid Dynamics, July 1115, Istanbul, Turkey, 2016.
[ bib 
pdf
]
This is the origin of the development of highorder hyperbolic NavierStokes schemes based on reconstructed discontinuous Galekrin (rDG) methods,
which is a general framework including finitevolume and discontinuous Galerkin methods as special cases.
ActiveFlux:
Activeflux method is a compact highorder cellcentered FV scheme for unstructured grids.
The cellaveraged solution is updated by the flux that evolves independently at cell faces.
It achieves thirdorder 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.

Thirdorder timeaccurate scheme for advection diffusion
H. Nishikawa and P. L. Roe, Thirdorder activeflux scheme for advection diffusion: Hyperbolic diffusion, boundary condition, and Newton solver,
Computers and Fluids, 125, pp.7181 2016.
[ bib 
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journal 
slides 
seminar video 
slides2
]
ResidualDistribution (RD):
RD method allows the construction of secondorder 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).

Hyperbolic RD scheme for Dispersion
A FirstOrder Hyperbolic System Approach for Dispersion,
Journal of Computational Physics, Volume 321, 15 September 2016, Pages 593605.
[ bib 
pdf  journal ]
AIAA Paper 20161331, 54th AIAA Aerospace Sciences Meeting, 48 January, San Diego, California, 2016.
[pdf]

Shockcapturing hyperbolic RD schemes
Highorder shockcapturing hyperbolic residualdistribution schemes on irregular triangular grids, Computers and Fluids, Volume 131, 5 June 2016, Pages 2944, 2016.
[ bib 
journal ]
HighOrder ResidualDistribution Schemes for Discontinuous Problems on Irregular Triangular Grids, AIAA Paper 20161331, 54th AIAA Aerospace Sciences Meeting, 48 January, San Diego, California, 2016.
[ bib 
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]

A new design princicple for hyperbolic RD schemes
Improved secondorder hyperbolic residualdistribution scheme and its extension to thirdorder on arbitrary triangular grids, Journal of Computational Physics, 300, pp.455491, 2015.
[ bib 
pdf 
journal
]
HighOrder Hyperbolic ResidualDistribution Schemes on Arbitrary Triangular Grids, AIAA Paper 20152445,
22nd AIAA Computational Fluid Dynamics Conference, Dallas, 2015.
[ bib 
pdf
]

Highorder timeaccurate schemes.
Very efficient highorder hyperbolic schemes for timedependent advectiondiffusion problems: Third, fourth, and sixthorder, Computers and Fluids, 102, pp.131147 2014.
[ bib 
pdf 
journal 
slides ]

Extension to timeaccurate scheme.
FirstOrder Hyperbolic System Method for
TimeDependent AdvectionDiffusion Problems, NASATM2014218175, March 2014.
[ bib  pdf ]

First extension to advectiondiffusion, 2ndorder explicit.
H. Nishikawa, A FirstOrder System Approach for Diffusion Equation. II:
Unification of Advection and Diffusion,
Journal of Computational Physics, 227, pp. 39894016, 2010.
[ bib 
pdf 
journal ]

First paper of the hyperbolic method, 2ndorder explicit.
H. Nishikawa, A FirstOrder System Approach for Diffusion Equation. I: SecondOrder Residual Distribution Schemes,
Journal of Computational Physics, 227, pp. 315352, 2007.
[ bib 
pdf 
journal 
code ]
Others:
Other works related to the hyperbolic method.
[ Please let us know if we're missing any work. ]

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 20184166, AIAA 2018 Fluid Dynamics Conference, 25  29 June 2018, Atlanta, Georgia.
[
AIAA ]

Hyperbolic NavierStokes FiniteVolume Solver
Tsukasa Nagao, Atsushi Hashimoto, and Tetsuya Sato,
A Study on Time Evolution Method for Hyperbolic NavierStokes System,
2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 20180370) .
[
AIAA ]

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 20180175) .
[
AIAA ]

HighOrder Compact Schemes for Magnetizd Electron Fluid
Amareshwara Sainadh Chamarthi and Zhexu Wang and Rei Kawashima,
Weighted Nonlinear Schemes for Magnetized Electron Fluid in Quasineutral plasma,
2018 AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, (AIAA 20182194) .
[
AIAA ]

Hyperbolic NavierStokes FiiteVolume 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, GPPS170198, 2017, Shanghai, China.
[
pdf ]

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 624637.
[
Journal ]

Hyperbolic method for anisotropic diffusion problems.
Rei Kawashima, Kimiya Komurasaki, Tony Schönherra, A hyperbolicequation system approach for magnetized electron fluids in quasineutral plasmas, Journal of Computational Physics, Volume 284, Issue 1, March 2015, Pages 5969.
[
Journal ]

Entropy consitent Navier_Stokes methods via the hyperbolic method.
Akmal Nizam Mohammed and Farzad Ismail, Entropy Consistent Methods for the NavierStokes Equations,
Journal of Scientific Computing, August 2014, Pages 120.
[
Journal ]



