Hyperbolic Method

Hyperbolize, Discretize, and Solve.





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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 aplha-damping scheme, is now employed as a default scheme in a commercial CFD code [ Software Cradle's SC/Tetra ] .

  1. Effects of Damping Coefficient for Implicit Solver

    H. Nishikawa and Y. Nakashima and N. Watanabe, "Effects of High-Frequency Damping on Iterative Convergence of Implicit Viscous Solver", Journal of Computational Physics, Volume 348, November 2017, Pages 66-81
    [ bib | pdf | journal ]
  2. General recipe extended to Navier-Stokes

    H. Nishikawa, "Two Ways to Extend Diffusion Schemes to Navier-Stokes Schemes: Gradient Formula or Upwind Flux", AIAA Paper 2011-3044, 20th AIAA Computational Fluid Dynamics Conference, Hawaii, 2011.
    [ bib | pdf | slides ]
  3. A general recipe for constructing robust diffusion schemes

    Long Version
    H. Nishikawa, "Beyond Interface Gradient: A General Principle for Constructing Diffusion Schemes", AIAA Paper 2010-5093, 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.62-86 2011.
    [ bib | pdf | journal | note ]
  4. Originally discussed in Section 5 of the following paper

    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 ]




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).

  1. Gino I. Montecinos, Eleuterio F. Toro, "Reformulations for general advection-diffusion-reaction equations and locally implicit ADER schemes", Journal of Computational Physics, Volume 275, Issue 15, October 2014, Pages 415-442.
    [ journal ]
  2. 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 101-123.
    [ journal ]




Mixed Formulation:


Mixed formulation is different from the hyperbolic method in that the mixed formulation is not hyperbolic.

  1. Mixed formulation to ensure uniform accuracy for advection-diffusion equations in residual-distribution method

    H. Nishikawa and P. L. Roe, On High-Order Fluctuation-Splitting Schemes for Navier-Stokes Equations, Computational Fluid Dynamics 2004: Proceedings of the Third International Conference on Computational Fluid Dynamics, ICCFD, Toronto, 12-16 July 2004, Springer, 2006.
    [ bib | pdf ]





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