Convergence of Generalized Muscl Schemes

  • Stanley Osher


Semidiscrete generalizations of the second order extension of Godunov’s scheme, known as the MUSCL scheme, are constructed, starting with any three point “E” scheme. They are used to approximate scalar conservation laws in one space dimension. For convex conservation laws, each member of a wide class is proven to be a convergent approximation to the correct physical solution. Comparison with another class of high resolution convergent schemes is made.


Riemann Problem Entropy Solution Entropy Condition Entropy Inequality Flux Function 
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  1. [1]
    S. R. Chakravarthy and S. Osher, High resolution applications of the Osher upwind scheme for the Euler equations, Proc. AIAA Computational Fluid Dynamics Conference, Danvers, MA, 1983, pp. 363–372.Google Scholar
  2. [2]
    P. Colella, A direct EulerianMUSCLscheme for gas dynamics, Lawrence Berkeley Lab. Report #LBL-14104, 1982.Google Scholar
  3. [3]
    R. J. Diperna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal., 82 (1983), pp. 27–70.MathSciNetADSMATHCrossRefGoogle Scholar
  4. [4]
    B. Engquist, and S. Osher, Stable and entropy condition satisfying approximations for transonic flow calculations, Math. Comp., 34 (1980), pp. 45–75.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    S. K. Godunov, A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics, Mat. Sb., 47 (1959), pp. 271 – 290.MathSciNetGoogle Scholar
  6. [6]
    B. Gustafsson, The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comp., 29 (1975), pp. 396 – 406.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    A. Harten High resolution schemes for hyperbolic conservation laws, J. Comp. Phys., 49 (1983), pp. 357–393.MathSciNetADSMATHCrossRefGoogle Scholar
  8. [8]
    A. Harten, On second order accurate Godunov-type schemes NASA AMES Report # NCA2-ORS25-201.Google Scholar
  9. [9]
    S. N. Kruzkov, First order quasi-linear equations in several independent variables Math. USSR Sb., 10 (1970), pp. 217–243.CrossRefGoogle Scholar
  10. [10]
    W. A. Mulder and B. Van Leer Implicit upwind computations for the Euler equations, AIAA Computational Fluid Dynamics Conference, Danvers, MA, 1983, pp. 303–310.Google Scholar
  11. [11]
    S. Osher, Riemann solvers, the entropy condition, and difference approximations, this Journal, 21 (1984), pp. 217 – 235.MathSciNetMATHGoogle Scholar
  12. [12]
    S. Osher and S. R. Chakravarthy High resolution schemes and the entropy condition, this Journal, 21 (1984), pp. 955 – 984.MathSciNetMATHGoogle Scholar
  13. [13]
    P. K. Sweby High resolution schemes using flux limiters for hyperbolic conservation laws, this Journal, 21 (1984), pp. 995–1011.MathSciNetMATHGoogle Scholar
  14. [14]
    P. K. Sweby and M. J. Baines Convergence of Roe’s scheme for the general non-linear scalar wave equation Numerical Analysis Report, 8/81, Univ. Reading, 1981.Google Scholar
  15. E. Tadmor Numerical viscosity and the entropy condition for conservative difference schemes ICASE NASA Contractor Report 172141, (1983), NASA Langley Research Center, Hampton, VA.Google Scholar
  16. [16]
    B. van Leer Towards the ultimate conservative difference scheme, II. Monotonicity and conservation combined in a second order scheme, J. Comp. Phys.,14 (1974), pp.361–376.ADSCrossRefGoogle Scholar
  17. [17]
    B. van Leer, Towards the ultimate conservative difference scheme. V. A. second-order sequel to Godunov’s method, J. Comp. Phys., 32 (1979), pp. 101–136.ADSCrossRefGoogle Scholar
  18. [18]
    H. C. Yee, R. F. Warming and A. Harten Implicit total variation diminishing (TVD) schemes for steady state calculations, Proc. AIAA Computational Fluid Dynamics Conference, Danvers, MA, 1983, pp. 110–127.Google Scholar
  19. [19]
    R. Sanders On convergence of monotone finite difference schemes with variable spatial differencing, Math. Comp.,40 (1983), pp. 91–106.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Stanley Osher
    • 1
  1. 1.Department of MathematicsUniversity of California at Los AngelesLos AngelesUSA

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