[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

On Flux-Difference Residual Distribution Methods

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

In this paper, the properties of the newly developed flux-difference residual distribution methods will be analyzed. The focus would be on the order-of-accuracy and stability variations with respect to changes in grid skewness. Overall, the accuracy loss and the stability range of the new methods are comparable with the existing residual distribution methods. It will also be shown that new method has a general mathematical formulation which can easily recover the existing residual distribution methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Abgrall, R.: Residual distribution schemes: current status and future trends. Comput. Fluids 35(7), 641–669 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abgrall, R.: A review of residual distribution schemes for hyperbolic and parabolic problems: the July 2010 state of the art. Commun. Comput. Phys. 11(4), 1043–1080 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abgrall, R., Trefilík, J.: An example of high order residual distribution scheme using non-Lagrange elements. J. Sci. Comput. 45(1–3), 3–25 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chizari, H., Ismail, F.: Accuracy variations in residual distribution and finite volume methods on triangular grids. Bull. Malays. Math. Sci. Soc. 22, 1–34 (2015)

    MATH  Google Scholar 

  5. Chizari, H., Ismail, F.: A grid-insensitive lda method on triangular grids solving the system of euler equations. J. Sci. Comput. 71(2), 839–874 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Courant, R., Friedrichs, K., Lewy, H.: On the partial difference equations of mathematical physics. IBM J. 11, 215–234 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  7. Deconinck, H., Roe, P., Struijs, R.: A multidimensional generalization of Roe’s flux difference splitter for the Euler equations. Comput. Fluids 22(2), 215–222 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ganesan, M.: Analytical study of residual distribution methods for solving conservation laws, MSc. thesis, Universiti Sains Malaysia (2017)

  9. Guzik, S., Groth, C.: Comparison of solution accuracy of multidimensional residual distribution and Godunov-type finite-volume methods. Int. J. Comput. Fluid Dyn. 22, 61–83 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ismail, F., Carrica, P.M., Xing, T., Stern, F.: Evaluation of linear and nonlinear convection schemes on multidimensional non-orthogonal grids with applications to KVLCC2 tanker. Int. J. Numer. Methods Fluids 64(September 2009), 850–886 (2010)

    MathSciNet  MATH  Google Scholar 

  11. Ismail, F., Chizari, H.: Developments of entropy-stable residual distribution methods for conservation laws I: scalar problems. J. Comput. Phys. 330, 1093–1115 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  12. Jameson, A.: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence in transonic and hypersonic flow. In: 93-3359. AIAA Conference, Orlando (1993)

  13. Katz, A., Sankaran, V.: High aspect ratio grid effects on the accuracy of Navier–Stokes solutions on unstructured meshes. Comput. Fluids 65, 66–79 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Masatsuka, K.: I do like CFD, book, vol. 1 (2009)

  15. Mazaheri, A., Nishikawa, H.: Improved second-order hyperbolic residual-distribution scheme and its extension to third-order on arbitrary triangular grids. J. Comput. Phys. 300(1), 455–491 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Nishikawa, H.: Fluctuation-Splitting Schemes and Hyp / Ell Decompositions of the Euler Fluctuations and Forms of The Euler Equa- tions. Tech. Rep. August (2004)

  17. Nishikawa, H.: A first-order system approach for diffusion equation. II: unification of advection–diffusion. J. Comput. Phys. 229(11), 3989–4016 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ricchiuto, M., Abgrall, R.: Explicit Runge–Kutta residual distribution schemes for time dependent problems: second order case. J. Comput. Phys. 229(16), 5653–5691 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Singh, V., Chizari, H., Ismail, F.: Non-unified Compact Residual-Distribution Methods for Scalar Advection-Diffusion Problems. Journal of Scientific Computing revision stage (2017)

Download references

Acknowledgements

We would like to thank the Ministry of Higher Education of Malaysia for financially supporting this research work under the FRGS grant (NO: 203/PAERO/6071316).

Author information

Authors and Affiliations

  1. School of Aerospace Engineering, Engineering Campus, Universiti Sains Malaysia, 14300, Pulau Pinang, Malaysia

    Farzad Ismail & Wei Shyang Chang

  2. State Key Laboratory for Strength and Vibration of Mechanical Structures, Xian Jiaotong University, Xi’an, 710049, People’s Republic of China

    Hossain Chizari

Authors
  1. Farzad Ismail
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Shyang Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hossain Chizari
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Farzad Ismail.

Additional information

Communicated by Ahmad Izani.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (txt 1 KB)

Appendices

A Truncation Error (TE) of Classic RD Methods

The TE for different classic RD methods are determined with the assumption that \(\frac{b}{a}<\frac{k}{h}\).

$$\begin{aligned} \text {TE}_\text {N}= & {} \left( -\frac{ab}{2r^2}(ak-bh)\right) u_{nn}\nonumber \\&+\,\left( \frac{ab}{6r^3}(ak-bh)(ak-2bh)\right) u_{nnn}\nonumber \\&+\,O(h^pk^q),\qquad p+q=3. \end{aligned}$$
(67)
$$\begin{aligned} \text {TE}_\text {LxF}= & {} \left( -\frac{\left( 4 a^3 k^3-3 a^2 b k^2h+4 a b^2 kh^2+b^3h^3\right) }{6kh\left( a^2+b^2\right) }\right) u_{nn}\nonumber \\&+\,\left( \frac{a b^2 (ak-bh)h}{12 \left( a^2+b^2\right) ^{3/2}}\right) u_{nnn}\nonumber \\&+\,O(h^pk^q),\qquad p+q=3. \end{aligned}$$
(68)
$$\begin{aligned} \text {TE}_\text {LDA}= & {} \left( \frac{ab(ak-bh)(ak-2bh)}{6r^3}\right) u_{nnn}\nonumber \\&+\,\left( -\frac{ab^2(ak-bh)^2h}{8r^4}\right) u_{nnnn}\nonumber \\&+\,O(h^pk^q),\qquad p+q=4. \end{aligned}$$
(69)
$$\begin{aligned} \text {TE}_\text {LxW}= & {} \left( -\frac{a b^2 (ak-bh)^2h}{12r^4}\right) u_{nnnn}\nonumber \\&+\,O(h^pk^q),\qquad p+q=4. \end{aligned}$$
(70)

B Stability Analysis on Classic RD Methods

With the same formulation in Sect. 3.6, the amplification factor, \(\delta \) for N and LDA schemes are determined as the following.

$$\begin{aligned} |\delta |^{\mathrm{N}}= & {} \left\{ \left[ 1 + \frac{\nu }{3 s} \left( -s +(s-1) \hbox {cos}(\theta x) + \hbox {cos}(\theta x + \theta y)\right) \right] ^2\right. \nonumber \\&+ \left. \left[ i \frac{\nu }{3 s} \left( (s -1)\hbox {sin}(\theta x) + \hbox {sin}(\theta x + \theta y)\right) \right] ^2\right\} ^{\frac{1}{2}}. \end{aligned}$$
(71)
$$\begin{aligned} |\delta |^{\mathrm{LDA}}= & {} \left\{ \left[ 1 + \frac{\nu }{3 s^2} \left( -s^2+s-1+(s-1)s \hbox {cos}(\theta x)\right. \right. \right. \nonumber \\&\left. \left. +(1-s)\hbox {cos}(\theta y)+ s \hbox {cos}(\theta x + \theta y)\right) \right] ^2\nonumber \\&+ \left. \left[ i \frac{\nu }{3 s} \left( (s -1)\hbox {sin}(\theta x) + \hbox {sin}(\theta x + \theta y)\right) \right] ^2\right\} ^{\frac{1}{2}}. \end{aligned}$$
(72)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ismail, F., Chang, W.S. & Chizari, H. On Flux-Difference Residual Distribution Methods. Bull. Malays. Math. Sci. Soc. 41, 1629–1655 (2018). https://doi.org/10.1007/s40840-017-0559-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-017-0559-8

Keywords

Mathematics Subject Classification

Search

Navigation