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History
2007: Diffusion
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.
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The idea of using the hyperbolic diffusion system for steady computations was first introduced in this paper.
The key idea is that the hyperbolic system recovers the diffusion
equation in the steady state.
The hyperbolic system is in the same form as Cattaneo's hyperbolic heat equation.
However, the relaxation time is determined not by physical consideration, but for
fast steady convergence. Consequently, the relaxation time is arbitrary and does not have to be
small, thus leading to a non-stiff formulation.
2010: Advection Diffusion
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.
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The hyperbolic diffusion is extended to advection-diffusion in this paper.
The extension is unique.
Unlike other relaxation methods, the hyperbolic advection-diffusion system
is constructed by adding an advective term to the hyperbolic diffusion system introduced
in the previous paper. This type of construction is not found in other relaxation-type methods.
This paper shows that the advective term and the diffusive term can be integrated into a single
hyperbolic system, and a scheme developed for hyperbolic systems applies directly to the integrated system:
no need to devise a scheme for diffusion and add it to an advection scheme.
Also, the relaxation time is now determined not specifically for a numerical scheme, but
more generally for the differential system, again for fast steady convergence.
2011: Navier-Stokes
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.
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A hyperbolic Navier-Stokes system is constructed and demonstrated by a second-order
finite-volume method on unstructured grids. In this paper, a separate treatment of the inviscid
and the hyperbolic viscous system was proposed as a practical way to construct a Navier-Stokes scheme.
This is partly because a full integration of the inviscid and the hyperbolic viscous system is difficult:
eigenvalues have not yet been found yet analytically for the whole system. But the hyperbolic viscous system
can be fully analyzed and proved to be hyperbolic. Second-order accuracy is demonstrated for
the viscous stresses and the heat fluxes on irregular grids at orders-of-magnitude acceleration in
convergence.
Since then, the method has been applied to various discretization methods such as
finite-volume, residual-distribution, and active-flux methods. It is applicable to virtually
any discretization method because the idea is to rewrite the governing equation as hyperbolic and
then discretize it.
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Hyperbolic Method