Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T18:25:04.556Z Has data issue: false hasContentIssue false
  • English
  • Français

On 16th and 32th Order Multioperators-Based Schemes for Smooth and Discontinuous Fluid Dynamics Solutions

Part of: Shock waves and blast waves Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  21 June 2017

Andrei I. Tolstykh
Andrei I. Tolstykh*
Affiliation:
Dorodnicyn Computing Center, Federal Research Center “Computer Science and Control” of Russian Academy of Sciences, 117999 Moscow GSP-1, Vavilova str. 40, Moscow Institute of Physics and Technology, Russia
*
*Corresponding author. Email address:[email protected] (A. I. Tolstykh)
Get access

Abstract

The paper presents a novel family of arbitrary high order multioperators approximations for convection, convection-diffusion or the fluid dynamics equations. As particular cases, the 16th- and 32th-order skew-symmetric multioperators for derivatives supplied by the 15th- and 31th-order dissipation multioperators are described. Their spectral properties and the comparative efficiency of the related schemes in the case of smooth solutions are outlined. The ability of the constructed conservative schemes to deal with discontinuous solutions is investigated. Several types of nonlinear hybrid schemes are suggested and tested against benchmark problems.

Keywords

Arbitrary-order discretizationmultioperatorsequations with convection terms16th- and 32th-order schemesdiscontinuous solutions

MSC classification

Secondary: 65M06: Finite difference methods 65M99: None of the above, but in this section 76L05: Shock waves and blast waves 76L99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 22 , Issue 2 , August 2017 , pp. 572 - 598
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Chi-Wang Shu

References

[1] Tolstykh, A. I., Multioperator high-order compact upwind methods for CFD parallel calculations, Parallel Computational Fluid Dynamics, Elsevier, Amsterdam (1998), 383–390.Google Scholar
[2] Tolstykh, A. I., Development of arbitrary-order multioperators-based schemes for parallel calculations 1, higher-than-fifth order approximations to convection terms, J. Comput. Phys., 225 (2007), 2333–2353.Google Scholar
[3] Lee, S. K., Compact finite difference schemes with spectral-like resolution. J. Comput. Phys., 103 (1992), 16–42.Google Scholar
[4] Shu, C.-W. and Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes II, J. Comput. Phys., 83 (1989), 32–78.CrossRefGoogle Scholar
[5] Liu, X. D., Osher, S. and Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115 (1994), 200–212.Google Scholar
[6] Balsara, D. S. and Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy, J. Comput. Phys., 160 (2000), 405–452.Google Scholar
[7] Titarev, V. A. and Toro, E. F., ADER schemes for three-dimensional nonlinear hyperbolic systems, J. Comput. Phys., 204 (2005), 715–736.Google Scholar
[8] Schwartzkopff, M. D. and Munz, C. D., Fast high order ADER schemes for linear hyperbolic equations, J. Comput. Phys., 197(2) (2004), 532–539.Google Scholar
[9] Tolstykh, A. I., Development of arbitrary-order multioperators-based schemes for parallel calculations 2, families of compact approximations with two-diagonal inversions and related multioperators, J. Comput. Phys., 227 (2008), 2922–2940.Google Scholar
[10] Tolstykh, A. I., High Accuracy Non-Centered Compact Difference Schemes for Fluid Dynamics Applications, World Scientific, Singapore, 1994.CrossRefGoogle Scholar
[11] Tolstykh, A. I. and Shirobokov, D. A., Fast calculations of screech using highly accurate multioperators-based schemes, J. Appl. Acoustics., 74 (2013), 102–109.Google Scholar
[12] Lipavskii, M. V. and Tolstykh, A. I., Tenth-order accurate multioperators scheme and its application in direct numerical simulation, J. Comput. Math. Math. Phys., 53(4) (2013), 455–468.CrossRefGoogle Scholar
[13] Tam, C. K. W., Problem 1-aliasing, Fourth Computational Aeroacoustics (CAA) Workshop on benchmark problems, NASA/CP-2004-212954, (2004).Google Scholar
[14] Adams, N. A. and Shariff, K., A high resolution Compact-ENO schemes for shock-turbulence interaction problems, J. Comput. Phys., 127 (1996), 27–51.Google Scholar
[15] Lax, P. D., Week solutions of nonlinear hyperbolic equations and their numerical cpmputations, Commun. Pure Appl. Math., 7(1) (1954), 159–193.CrossRefGoogle Scholar
[16] Tolstykh, A. I., On hybrid schemes with high-order multioperators for calculations of discontinuous solutions, Comput. Math. Math. Phys., 53(9) (2013), 1481–1502.Google Scholar
[17] Sod, G. A., A survey of several finite difference schemes for hyperbolic conservation laws, J. Comput. Phys., 27 (1978), 1–31.Google Scholar
[18] Woodward, P. and Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), 115–173.CrossRefGoogle Scholar
[19] Fedorenko, R. P., Using high-accuracy schemes for numerical solutions of hyperboloic equations, Comput. Math. Math. Phys., 2(6) (1962), 1122–1128.Google Scholar
[20] Petrov, I. B. and Kholodov, A. S., On regularization of discontinuous numerical solutions of hyperbolic equations, Comput. Math. Math. Phys., 24(8) (1984), 1172–1188.Google Scholar
[21] Mikhailovskaya, M.N. and Rogov, B. V., Monotone compact schemes for hyperbolic systems, Comput. Math. Math. Phys., 52(4) (2012), 672–695.Google Scholar
[22] Zalesac, S. T., Fully multidimensionalflux-corrected transport algorithms for fluids, J. Comput. Phys., 31 (1979), 335–362 Google Scholar
[23] Boris, J. P. and Book, D. L. , Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11 (1973), 38–69.Google Scholar
[24] Toro, R., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin, Heidelberg, 1997.Google Scholar
[25] Liska, R. and Wendroff, B., Comparison of several difference schemes on1D and 2D test problems for the Euler equations, SIAM J. Sci. Comput., 26 (2003), 995–1017.Google Scholar
[26] Noh, W. F., Errors for calculations of strong shocks using an artificial viscosity and artificial heat flux, J. Comput. Phys., 72 (1987), 78–120.Google Scholar
[27] Shu, C. W. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes II, J. Comput. Phys., 83 (1989), 32–78 Google Scholar