

Related Topics
Note: The following methods are related but different from the hyperbolic method considered in this website.
Construction of Diffusion/Viscous Schemes:
Hyperbolic method can be used as a construction method for a conventional diffusion/viscous scheme.
First, discretize the hyperbolized system by upwind schemes,
and then discard extra equations and reconstruct the extra variables (e.g., gradients and viscous stresses) from the main variables.
The resulting scheme is a robust and accurate schemes for diffusion. Note: These schemes do not require extra variables and
equations.
One of the schemes derived from this study, called the aplhadamping scheme, is now employed
as a default scheme in a commercial CFD code [
Software Cradle's SC/Tetra ] .

Effects of Damping Coefficient for Implicit Solver
H. Nishikawa and Y. Nakashima and N. Watanabe,
"Effects of HighFrequency Damping on Iterative Convergence of Implicit Viscous Solver",
Journal of Computational Physics, Volume 348, November 2017, Pages 6681
[ bib 
pdf 
journal
]
 General recipe extended to NavierStokes
H. Nishikawa, "Two Ways to Extend Diffusion Schemes to
NavierStokes Schemes: Gradient Formula or Upwind Flux", AIAA Paper 20113044,
20th AIAA Computational Fluid Dynamics Conference, Hawaii, 2011.
[ bib 
pdf 
slides ]

A general recipe for constructing robust diffusion schemes
Long Version
H. Nishikawa, "Beyond Interface Gradient: A General Principle for Constructing Diffusion Schemes",
AIAA Paper 20105093, 40th AIAA Fluid Dynamics Conference and Exhibit,
Chicago, 2010.
[ bib 
pdf 
slides
note ]
Short Version
H. Nishikawa, "Robust and Accurate Viscous Discretization via Upwind Scheme  I:
Basic Principle", Computers and Fluids, 49, pp.6286 2011.
[ bib 
pdf 
journal 
note ]
 Originally discussed in Section 5 of the following paper
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 ]
Relaxation methods:
These methods are different from the hyperbolic method considered in this website in that
the relaxation time is very small, thus leading to a stiff hyperbolic system or to
the usual diffusion restriction O(h^2) for diffusion. The following papers are especially relevant
in that the hyperbolic formulation is very similar to the one employed in the
hyperbolic method; but the relaxation time is quite different (dependent on the mesh size).
 Gino I. Montecinos, Eleuterio F. Toro,
"Reformulations for general advectiondiffusionreaction equations and locally implicit ADER schemes",
Journal of Computational Physics, Volume 275, Issue 15, October 2014, Pages 415442.
[ journal ]
 Gino I. Montecinos, Lucas O. Müller, Eleuterio F. Toro,
"Hyperbolic reformulation of a 1D viscoelastic blood flow model and ADER finite volume schemes",
Journal of Computational Physics, Volume 266, 1, June 2014, Pages 101123.
[ journal ]
Mixed Formulation:
Mixed formulation is different from the hyperbolic method in that the mixed
formulation is not hyperbolic.
 Mixed formulation to ensure uniform accuracy for advectiondiffusion equations
in residualdistribution method
H. Nishikawa and P. L. Roe, On HighOrder FluctuationSplitting Schemes
for NavierStokes Equations, Computational Fluid Dynamics 2004: Proceedings of the Third International Conference on Computational Fluid
Dynamics, ICCFD, Toronto, 1216 July 2004, Springer, 2006.
[ bib  pdf ]



