research-article

Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations

Authors:

Burton WendroffAuthors Info & Claims

SIAM Journal on Scientific Computing, Volume 25, Issue 3

Pages 995 - 1017

https://doi.org/10.1137/S1064827502402120

Published: 01 January 2003 Publication History

Abstract

The results of computations with eight explicit finite difference schemes on a suite of one-dimensional and two-dimensional test problems for the Euler equations are presented in various formats. Both dimensionally split and two-dimensional schemes are represented, as are central and upwind-biased methods, and all are at least second-order accurate.

References

[1]

M. Abouziarov, On nonlinear amplification mechanism of instability near density discontinuities and shock waves for finite volume methods for Euler equations, in HYP 2002, The Ninth International Conference on Hyperbolic Problems: Theory, Numerics and Applications, Abstracts, T. Hou and E. Tadmor, eds., Caltech, Pasadena, CA, 2002, p. 162.

[2]

P. Arminjon, D. Stanescu, and M. Viallon, A two‐dimensional finite volume extension of the Lax‐Friedrichs and Nessyahy‐Tadmor schemes for compressible flows, in Proceedings of the 6th International Symposium on Computational Fluid Dynamics, Vol. 4, M. Hafez and K. Oshima, eds., Lake Tahoe, NV, 1995, pp. 7–14.

[3]

T. Aslam, personal communication, 2001.

[4]

S. Billett, E. Toro, On WAF‐type schemes for multidimensional hyperbolic conservation laws, J. Comput. Phys., 130 (1997), 1–24

Digital Library

[5]

J. Blondin, VH‐1: The Virginia Numerical Bull Session Ideal Hydrodynamics PPMLR, http://wonka.physics.ncsu.edu/pub/VH‐1/index.html http://wonka.physics.ncsu.edu/pub/VH‐1/index.html.

[6]

E. Caramana, D. Burton, M. Shashkov, P. Whalen, The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys., 146 (1998), 227–262

Digital Library

[7]

Sin‐Chung Chang, Xiao‐Yen Wang, Chuen‐Yen Chow, The space‐time conservation element and solution element method: a new high‐resolution and genuinely multidimensional paradigm for solving conservation laws, J. Comput. Phys., 156 (1999), 89–136

Digital Library

[8]

P. Colella and P. Woodward, The piecewise parabolic method (PPM) for gas‐dynamical simulations, J. Comput. Phys., 54 (1984), pp. 174–201.

[9]

Constantine Dafermos, Hyperbolic conservation laws in continuum physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 325, Springer‐Verlag, 2000xvi+443

[10]

Edwige Godlewski, Pierre‐Arnaud Raviart, Numerical approximation of hyperbolic systems of conservation laws, Applied Mathematical Sciences, Vol. 118, Springer‐Verlag, 1996viii+509

[11]

P. Gresho, On the theory of semi‐implicit projection methods for viscous incompressible flow and its implementation via finite‐element method that also introduces a nearly consistent mass matrix: Part 2: Applications, Int. J. Numer. Methods Fluids, 11 (1990), pp. 621–659.

[12]

J. Gressier and J. M. Moschetta, Robustness versus accuracy in shock‐wave computations, Int. J. Numer. Methods Fluids, 33 (2000), pp. 313–332.

[13]

J. Grove and V. Mousseau, personal communication, Los Alamos National Laboratory, Los Alamos, NM, 2000.

[14]

H. Hanche‐Olsen, H. Holden, and K. A. Lie, Conservation Laws Preprint Server,http://www.math.ntnu.no/conservation http://www.math.ntnu.no/conservation.

[15]

A. Harten, The artificial compression method for computation of shocks and contact discontinuities: III. Self adjusting hybrid schemes, Math. Comp., 32 (1978), pp. 363–389.

[16]

Ami Harten, Björn Engquist, Stanley Osher, Sukumar Chakravarthy, Uniformly high‐order accurate essentially nonoscillatory schemes. III, J. Comput. Phys., 71 (1987), 231–303

Digital Library

[17]

W. Hui, P. Li, Z. Li, A unified coordinate system for solving the two‐dimensional Euler equations, J. Comput. Phys., 153 (1999), 596–637

Digital Library

[18]

Guang‐Shan Jiang, Eitan Tadmor, Nonoscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., 19 (1998), 1892–1917

Digital Library

[19]

Guang‐Shan Jiang, Chi‐Wang Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126 (1996), 202–228

Digital Library

[20]

Shi Jin, Zhou Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math., 48 (1995), 235–276

[21]

Dietmar Kröner, Mario Ohlberger, A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions, Math. Comp., 69 (2000), 25–39

Digital Library

[22]

M. Kuchařík, personal communication, 2001.

[23]

A. Kurganov and E. Tadmor, Solution of Two‐Dimensional Riemann Problems for Gas Dynamics without Riemann Problem Solvers, Technical report CAM 00‐34, UCLA, Los Angeles, 2000.

[24]

Culbert Laney, Computational gasdynamics, Cambridge University Press, 1998xiv+613

[25]

Peter Lax, Xu‐Dong Liu, Solution of two‐dimensional Riemann problems of gas dynamics by positive schemes, SIAM J. Sci. Comput., 19 (1998), 319–340

Digital Library

[26]

Randall LeVeque, Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, 1990x+214

[27]

Randall Leveque, High‐resolution conservative algorithms for advection in incompressible flow, SIAM J. Numer. Anal., 33 (1996), 627–665

Digital Library

[28]

R. LeVeque, Wave propagation algorithms for multi‐dimensional hyperbolic systems, J. Comput. Phys., 131 (1997), pp. 327–353.

Digital Library

[29]

R. LeVeque, Clawpack Version 4.0 User’s Guide, Technical report, University of Washington, Seattle, 1999. Available online at http://www.amath.washington.edu

\tilde{/}

claw/ http://www.amath.washington.edu/ claw/.

[30]

Richard Liska, Burton Wendroff, Composite schemes for conservation laws, SIAM J. Numer. Anal., 35 (1998), 2250–2271

Digital Library

[31]

R. Liska and B. Wendroff, Comparison of Several Difference Schemes on 1d and 2d Test Problems for the Euler Equations, Technical report, Czech Technical University, Prague, 2002. Available online with animations at http://www‐troja.fjfi.cvut.cz

\tilde{/}

liska/CompareEuler/compare8.

[32]

X. Liu, personal communication, 2001.

[33]

X. Liu and P. Lax, Positive schemes for solving multi‐dimensional hyperbolic systems of conservation laws, Computational Fluid Dynamics Journal, 5 (1996), pp. 1–24.Available online at http://www.math.ucsb.edu

\tilde{/}

xliu/publication/paper/positive1.pdf.

[34]

K. Morton, On the analysis of finite volume methods for evolutionary problems, SIAM J. Numer. Anal., 35 (1998), 2195–2222

Digital Library

[35]

Haim Nessyahu, Eitan Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408–463

Digital Library

[36]

W. F. Noh, Errors for calculations of strong shocks using anartificial viscosity and artificial heat flux, J. Comput. Phys., 72 (1987), pp. 78–120.

Digital Library

[37]

J. Quirk, An Adaptive Grid Algorithm for Computational Shock Hydrodynamics, Ph.D. thesis, College of Aeronautics, Cranfield University, Cranfield, UK, 1991. Available online at http://www.galcit.caltech.edu

\tilde{/}

jjq/doc/am

\underline{r}

sol/thesis.

[38]

James Quirk, A contribution to the great Riemann solver debate, Internat. J. Numer. Methods Fluids, 18 (1994), 555–574

[39]

J. Quirk, personal communication, 2001.

[40]

W. J. Rider, Revisiting wall heating, J. Comput. Phys., 162 (2000), pp. 395–410.

Digital Library

[41]

Patrick Roache, Computational fluid dynamics, Hermosa Publishers, Albuquerque, N.M., 1976vii+446, With an appendix (“On artificial viscosity”) reprinted from J. Computational Phys. 10 (1972), no. 2, 169–184; Revised printing

[42]

M. S. Sahota, P. J. O’Rourke, and M. C. Cline, Assesment of Vorticity and Angular Momentum Errors in CHAD for the Gresho Problem, Technical report LA‐UR‐00‐2217, Los Alamos National Laboratory, Los Alamos, NM, 2000.

[43]

Carsten Schulz‐Rinne, James Collins, Harland Glaz, Numerical solution of the Riemann problem for two‐dimensional gas dynamics, SIAM J. Sci. Comput., 14 (1993), 1394–1414

Digital Library

[44]

Chi‐Wang Shu, Stanley Osher, Efficient implementation of essentially nonoscillatory shock‐capturing schemes. II, J. Comput. Phys., 83 (1989), 32–78

[45]

Chi‐Wang Shu, Stanley Osher, Efficient implementation of essentially nonoscillatory shock‐capturing schemes, J. Comput. Phys., 77 (1988), 439–471

Digital Library

[46]

Joel Smoller, Shock waves and reaction‐diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 258, Springer‐Verlag, 1994xxiv+632

[47]

G. A. Sod, A survey of several finite difference schemes for hyperbolic conservation laws, J. Comput. Phys., 27 (1978), pp. 1–31.

[48]

G. Strang, On the construction and comparison of difference schemes, SIAM J. Numer. Anal., 5 (1968), pp. 506–517.

Digital Library

[49]

E. Toro, A weighted average flux method for hyperbolic conservation laws, Proc. Roy. Soc. London Ser. A, 423 (1989), p. 401–418.

[50]

Eleuterio Toro, Riemann solvers and numerical methods for fluid dynamics, Springer‐Verlag, 1997xviii+592, A practical introduction

[51]

E. Toro, Numerica: A Library of Source Codes for Teaching, Research and Applications, 2000.

Available online at http://www.numeritek.com/numeric

\underline{a}

software.html.

[52]

Paul Woodward, Phillip Colella, The numerical simulation of two‐dimensional fluid flow with strong shocks, J. Comput. Phys., 54 (1984), 115–173

[53]

P. Woodward, S. E. Anderson, D. H. Porter, I. Quesnell, M. Jacobs, and B. Edgar, PPMLib Library, University of Minnesota, Minneapolis, 1999.

Available online at http://www.lcse.umn.edu/PPMlib/.

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Index Terms

Comparison of Several Difference Schemes on 1D and 2D Test Problems for the Euler Equations
1. Applied computing
  1. Physical sciences and engineering
2. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations
    2. Numerical analysis
      1. Computations in finite fields
      2. Numerical differentiation

Index terms have been assigned to the content through auto-classification.

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cover image SIAM Journal on Scientific Computing

SIAM Journal on Scientific Computing Volume 25, Issue 3

2003

351 pages

ISSN:1064-8275

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Copyright © 2003 Society for Industrial and Applied Mathematics.

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Society for Industrial and Applied Mathematics

United States

Publication History

Published: 01 January 2003

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Feng YWinter JAdams NSchranner F(2024)A general multi-objective Bayesian optimization framework for the design of hybrid schemes towards adaptive complex flow simulationsJournal of Computational Physics10.1016/j.jcp.2024.113088510:COnline publication date: 1-Aug-2024
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