Skip to main content

Approximate solutions of nonlinear conservation laws

Part of the Lecture Notes in Mathematics book series (LNMCIME,volume 1697)

Abstract

This is a summary of five lectures delivered at the CIME course on “Advanced Numerical Approximation of Nonlinear Hyperbolic Equations” held in Cetraro, Italy, on June 1997.

Following the introductory lecture I—which provides a general overview of approximate solution to nonlinear conservation laws, the remaining lectures deal with the specifics of four complementing topics:

  • -Lecture II. Finite-difference methods-non-oscillatory central schemes;

  • -Lecture III. Spectral approximations-the Spectral Viscosity method;

  • -Lecture IV. Convergence rate estimates-a Lip' convergence theory;

  • -Lecture V. Kinetic approximations-regularity of kinetic formulations.

MSC Subject Classification (1991)

  • Primary 35L65, 35L60
  • Secondary 65M06, 65M12, 65M15, 65M60, 65M70

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   69.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   89.99
Price excludes VAT (Canada)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Alvarez and J.-M. MorelFormulation and computational aspects of image analysis, Acta Numerica (1994), 1–59.

    Google Scholar 

  2. C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations II: convergence proofs of the Boltzmann equations, Comm. Pure Appl. Math. XLVI (1993), 667–754.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. G. Barles and P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asympt. Anal. 4 (1991), 271–283.

    MathSciNet  MATH  Google Scholar 

  4. J.B. Bell, P. Colella, and H.M. Glaz, A Second-Order Projection Method for the Incompressible Navier-Stokes Equations, J. Comp. Phys. 85 (1989), 257–283.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. M. Ben-Artzi and J. Falcovitz, Recent developments of the GRP method, JSME (Ser. b) 38 (1995), 497–517.

    Google Scholar 

  6. J.P. Boris and D. L. Book, Flux corrected transport: I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys. 11 (1973), 38–69.

    CrossRef  MATH  Google Scholar 

  7. F. Bouchut and B. PerthameKružkov's estimates for scalar conservation laws revisited, Universite D'Orleans, preprint, 1996.

    Google Scholar 

  8. F. Bouchot, CH. Bourdarias and B. Perthame, A MUSCL method satisfying all the numerical entropy inequalities, Math. Comp. 65 (1996) 1439–1461.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Y. Brenier, Résolution d'équations d'évolution quasilinéaires en dimension N d'espace à l'aide d'équations linéaires en dimension N+1, J. Diff. Eq. 50 (1983), 375–390.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Y. Brenier and S.J. Osher, The discrete one-sided Lipschitz condition for convex scalar conservation laws, SIAM J. Numer. Anal. 25 (1988), 8–23.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. A. Bressan, The semigroup approach to systems of conservation laws, 4th Workshop on PDEs, Part I (Rio de Janeiro, 1995). Mat. Contemp. 10 (1996), 21–74.

    MathSciNet  MATH  Google Scholar 

  12. A. Bressan, Decay and structural stability for solutions of nonlinear systems of conservation laws, 1st Euro-Conference on Hyperbolic Conservation Laws, Lyon, Feb. 1997.

    Google Scholar 

  13. A. Bressan and R. ColomboThe semigroup generated by 2×2 conservation laws, Arch. Rational Mech. Anal. 133 (1995), no. 1, 1–75.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. A. Bressan and R. ColomboUnique solutions of 2×2 conservation laws with large data, Indiana Univ. Math. J. 44 (1995), no. 3, 677–725.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. C. Cercignani, The Boltzmann Equation and its Applications, Appl. Mathematical Sci. 67, Springer, New-York, 1988.

    MATH  Google Scholar 

  16. T. Chang and L. Hsiao, The Riemann Problem and Interaction of Waves in Gasdynamics, Pitman monographs and surveys in pure appl. math, 41, John Wiley, 1989.

    Google Scholar 

  17. G.-Q. Chen, The theory of compensated compactness and the system of isentropic gas dynamics, Preprint MCS-P154-0590, Univ. of Chicago, 1990.

    Google Scholar 

  18. G.-Q. Chen, Q. Du and E. Tadmor, Spectral viscosity approximation to multidimensional scalar conservation laws, Math. of Comp. 57 (1993).

    Google Scholar 

  19. G.-Q. Chen, D. Levermore and T. P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy Comm. Pure Appl. Math. 47 (1994) 787–830.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. I. L. Chern, Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities, Comm. Pure Appl. Math. XLII (1989), 815–844.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. I.L. Chern, J. Glimm, O. McBryan, B. Plohr and S. Yaniv, Front Tracking for gas dynamics J. Comput. Phys. 62 (1986) 83–110.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. A. J. Chorin, Random choice solution of hyperbolic systems, J. Comp. Phys. 22 (1976), 517–533.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. B. Cockburn, Quasimonotone schemes for scalar conservation laws. I. II, III. SIAM J. Numer. Anal. 26 (1989) 1325–1341, 27 (1990) 247–258, 259–276.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. B. Cockburn, F. Coquel and P. LeFloch, Convergence of finite volume methods for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), 687–705.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. P. Colella, Multidimensional upwind methods for hyperbolic conservation laws, J. Comput. Phys. 87 (1990), 87–171.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. P. Colella and P. Woodward, The piecewise parabolic method (PPM) for gas-dynamical simulations, JCP 54 (1984), 174–201.

    MathSciNet  MATH  Google Scholar 

  27. F. Coquel and P. LeFloch, Convergence of finite difference schemes for conservation laws in several space dimensions: a general theory, SIAM J. Numer. Anal. (1993)

    Google Scholar 

  28. M. G. Crandall, The semigroup approach to first order quasilinear equations in several space dimensions, Israel J. Math. 12 (1972), 108–132.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. M. G. Crandall and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Math. of Comp. 34 (1980), 1–21.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. M. G. Crandall and A. Majda, The method of fractional steps for conservation laws, Numer. Math. 34 (1980), 285–314.

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. M. G. Crandall and L. Tartar, Some relations between non expansive and order preserving mapping, Proc. Amer. Math. Soc. 78 (1980), 385–390.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. T. Chang and S. Yang, Two-Dimensional Riemann Problems for Systems of Conservation Laws, Pitman Monographs and Surveys in Pure and Appl. Math., 1995.

    Google Scholar 

  34. C. Dafermos, Polygonal approximations of solutions of initial-value problem for a conservation law, J. Math. Anal. Appl. 38 (1972) 33–41.

    CrossRef  MathSciNet  MATH  Google Scholar 

  35. C. Dafermos, Hyperbolic systems of conservation laws, in “Systems of Nonlinear PDEs”, J. M. Ball, ed, NATO ASI Series C, No. 111, Dordrecht, D. Reidel (1983), 25–70.

    Google Scholar 

  36. C. Dafermos, private communication.

    Google Scholar 

  37. R. DeVore & G. Lorentz, Constructive Approximation, Springer-Verlag, 1993.

    Google Scholar 

  38. R. DeVore and B. Lucier, High order regularity for conservation laws, Indiana Univ. Math. J. 39 (1990), 413–430.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. S. M. Deshpande, A second order accurate, kinetic-theory based, method for inviscid compressible flows, NASA Langley Tech. paper No. 2613, 1986.

    Google Scholar 

  40. R. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rat. Mech. Anal. 82 (1983), 27–70.

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. R. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), 1–30.

    CrossRef  MathSciNet  MATH  Google Scholar 

  42. R. DiPerna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 88 (1985), 223–270.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. R. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math. 130 (1989), 321–366.

    CrossRef  MathSciNet  MATH  Google Scholar 

  44. R. DiPerna and P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), 729–757.

    CrossRef  MathSciNet  MATH  Google Scholar 

  45. R. DiPerna, P. L. Lions and Y. Meyer, Lpregularity of velocity averages, Ann. I.H.P. Anal. Non Lin. 8(3-2-4) (1991), 271–287.

    MathSciNet  MATH  Google Scholar 

  46. W. E. Shu and C.-W. Shu, A numerical resolution study of high order essentially non-oscillatory schemes applied to incompressible flow J. Comp. Phys. 110, (1993), 39–46.

    MATH  Google Scholar 

  47. B. Engquist and S. J. OsherOne-sided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981) 321–351.

    CrossRef  MathSciNet  MATH  Google Scholar 

  48. B. Engquist, P. Lotstedt and B. Sjogreen, Nonlinear filters for efficient shock computation, Math. Comp. 52 (1989), 509–537.

    CrossRef  MathSciNet  MATH  Google Scholar 

  49. B. Engquist and O. Runborg, Multi-phase computations in geometrical optics, J. Comp. Appl. Math. 74 (1996) 175–192.

    CrossRef  MathSciNet  MATH  Google Scholar 

  50. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential equations, AMS Regional Conference Series in Math. 74, Providence R.I. 1990.

    Google Scholar 

  51. K. FriedrichsSymmetric hyperbolic linear differential equations, CPAM 7 (1954) 345.

    MathSciNet  MATH  Google Scholar 

  52. K. O. Friedrichs and P. D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. USA 68 (1971), 1686–1688.

    CrossRef  MathSciNet  MATH  Google Scholar 

  53. P. Gérard, Microlocal defect measures, Comm. PDE 16 (1991), 1761–1794.

    CrossRef  MathSciNet  MATH  Google Scholar 

  54. Y. Giga and T. Miyakawa, A kinetic construction of global solutions of first-order quasilinear equations, Duke Math. J. 50 (1983), 505–515.

    CrossRef  MathSciNet  MATH  Google Scholar 

  55. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715.

    CrossRef  MathSciNet  MATH  Google Scholar 

  56. J. Glimm and P. D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Amer. Math. Soc. Memoir 101, AMS Providence, 1970.

    Google Scholar 

  57. J. Glimm, B. Lindquist and Q. Zhang, Front tracking, oil reservoirs, engineering scale problems and mass conservation in Multidimensional Hyperbolic Problems and Computations, Proc. IMA workshop (1989) IMA vol. Math Appl 29 (J. Glimm and A. Majda Eds.), Springer-Verlag, New-York (1991), 123–139.

    CrossRef  Google Scholar 

  58. E. Godlewski and P.-A. Raviart, Hyperbolic Systems of Conservation Laws, Ellipses, Paris, 1991.

    MATH  Google Scholar 

  59. E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws, Springer, 1996.

    Google Scholar 

  60. S. K. Godunov, A difference scheme for numerical computation of discontinuous solutions of fluid dynamics, Mat. Sb. 47 (1959), 271–306.

    MathSciNet  MATH  Google Scholar 

  61. S. K. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk. SSSR 139 (1961), 521–523.

    MathSciNet  MATH  Google Scholar 

  62. F. Golse, P. L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. of Funct. Anal. 76 (1988), 110–125.

    CrossRef  MathSciNet  MATH  Google Scholar 

  63. J. Goodman, private communication.

    Google Scholar 

  64. J. Goodman and P. D. Lax, On dispersive difference schemes. I, Comm. Pure Appl. Math. 41 (1988), 591–613.

    CrossRef  MathSciNet  MATH  Google Scholar 

  65. J. Goodman and R. LeVeque, On the accuracy of stable schemes for 2D scalar conservation laws, Math. of Comp. 45 (1985), 15–21.

    MathSciNet  MATH  Google Scholar 

  66. J. Goodman and J. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws, Arch. Rat. Mech. Anal. 121 (1992), 235–265.

    CrossRef  MathSciNet  MATH  Google Scholar 

  67. D. Gottlieb and E. Tadmor, Recovering Pointwise Values of Discontinuous Data within Spectral Accuracy, in “Progress and Supercomputing in Computational Fluid Dynamics”, Progress in Scientific Computing, Vol. 6 (E. M. Murman and S. S. Abarbanel, eds.), Birkhauser, Boston, 1985, 357–375.

    CrossRef  Google Scholar 

  68. E. Harabetian, A convergent series expansion for hyperbolic systems of conservation laws, Trans. Amer. Math. Soc. 294 (1986), no. 2, 383–424.

    CrossRef  MathSciNet  MATH  Google Scholar 

  69. A. Harten, The artificial compression method for the computation of shocks and contact discontinuities: I. single conservation laws, CPAM 39 (1977), 611–638.

    MathSciNet  MATH  Google Scholar 

  70. A. Harten, High resolution schemes for hyperbolic conservation laws, J. Comput. Phys. 49 (1983), 357–393.

    CrossRef  MathSciNet  MATH  Google Scholar 

  71. A. Harten, B. Engquist, S. Osher and S. R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes. III, JCP 71, 1982, 231–303.

    MathSciNet  MATH  Google Scholar 

  72. A. Harten M. Hyman and P. Lax, On finite-difference approximations and entropy conditions for shocks, Comm. Pure Appl. Math. 29 (1976), 297–322.

    CrossRef  MathSciNet  MATH  Google Scholar 

  73. A. Harten P. D. Lax and B. van Leer, On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983), 35–61.

    CrossRef  MathSciNet  MATH  Google Scholar 

  74. A. Harten and S. Osher, Uniformly high order accurate non-oscillatory scheme. I, SIAM J. Numer. Anal. 24 (1982) 229–309.

    MathSciNet  Google Scholar 

  75. C. Hirsch, Numerical Computation of Internal and External Flows, Wiley, 1988.

    Google Scholar 

  76. T. J. R. Hughes and M. Mallet, A new finite element formulation for the computational fluid dynamics: III. The general streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Engrg. 58 (1986), 305–328.

    CrossRef  MathSciNet  MATH  Google Scholar 

  77. F. James, Y.-J. Peng and B. Perthame, Kinetic formulation for chromatography and some other hyperbolic systems, J. Math. Pures Appl. 74 (1995), 367–385.

    MathSciNet  MATH  Google Scholar 

  78. F. John, Partial Differential Equations, 4th ed. Springer, New-York, 1982.

    CrossRef  MATH  Google Scholar 

  79. C. Johnson and A. Szepessy, Convergence of a finite element methods for a nonlinear hyperbolic conservation law, Math. of Comp. 49 (1988), 427–444.

    CrossRef  MathSciNet  MATH  Google Scholar 

  80. C. Johnson, A. Szepessy and P. Hansbo, On the convergence of shockcapturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. of Comp. 54 (1990), 107–129.

    CrossRef  MathSciNet  MATH  Google Scholar 

  81. G.-S. Jiang and E. Tadmor, Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws, SIAM J. Sci. Compt., in press.

    Google Scholar 

  82. S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277.

    CrossRef  MathSciNet  MATH  Google Scholar 

  83. T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rat. Mech. Anal. 58 (1975), 181–205.

    CrossRef  MathSciNet  MATH  Google Scholar 

  84. Y. Kobayashi, An operator theoretic method for solving u t =δΨ(u), Hiroshima Math. J. 17 (1987) 79–89.

    MathSciNet  Google Scholar 

  85. D. Kröner, S. Noelle and M. Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions, Numer. Math. 71 (1995) 527–560.

    CrossRef  MathSciNet  MATH  Google Scholar 

  86. D. Kröner and M. Rokyta, Convergence of Upwind Finite Volume Schemes for Scalar Conservation Laws in two space dimensions, SINUM 31 (1994) 324–343.

    CrossRef  MATH  Google Scholar 

  87. S. N. Kružkov, The method of finite difference for a first order non-linear equation with many independent variables, USSR comput Math. and Math. Phys. 6 (1966), 136–151. (English Trans.)

    CrossRef  Google Scholar 

  88. S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR Sbornik 10 (1970), 217–243.

    CrossRef  Google Scholar 

  89. R. Kupferman and E. Tadmor, A fast high-resolution second-order central scheme for incompressible flows, Proc. Nat. Acad. Sci. 94 (1997), 4848–4852

    CrossRef  MathSciNet  MATH  Google Scholar 

  90. N. N. Kuznetsov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equations, USSR Comp. Math. and Math. Phys. 16 (1976), 105–119.

    CrossRef  Google Scholar 

  91. P. D. Lax, Weak solutions of non-linear hyperbolic equations and their numerical computations, Comm. Pure Appl. Math. 7 (1954), 159–193.

    CrossRef  MathSciNet  MATH  Google Scholar 

  92. P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math. 10 (1957), 537–566.

    CrossRef  MathSciNet  MATH  Google Scholar 

  93. P. D. Lax, Shock waves and entropy, in Contributions to nonlinear functional analysis, E. A. Zarantonello Ed., Academic Press, New-York (1971), 603–634.

    Google Scholar 

  94. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (SIAM, Philadelphia, 1973).

    CrossRef  MATH  Google Scholar 

  95. P. D. Lax and X.-D. Liu, Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws, Courant Mathematics and Computing Laboratory Report NYU, 95-003 (1995), Comm. Pure Appl. Math.

    Google Scholar 

  96. P. Lax and B. Wendroff, Systems of conservation laws, Comm. Pure Appl. Math. 13 (1960), 217–237.

    CrossRef  MathSciNet  MATH  Google Scholar 

  97. P. Lax and B. Wendroff, Difference schemes for hyperbolic equations with high order of accuracy, Comm. Pure Appl. Math. 17 (1964), 381.

    CrossRef  MathSciNet  MATH  Google Scholar 

  98. B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov's method, J. Comput. Phys. 32 (1979), 101–136.

    CrossRef  Google Scholar 

  99. F. LeFloch and P. A. Raviart, An asymptotic expansion for the solution of the generalized Riemann problem, Part I: General theory, Ann. Inst. H. Poincare, Nonlinear Analysis 5 (1988), 179.

    MathSciNet  Google Scholar 

  100. P. LeFloch and Z. Xin, Uniqueness via the adjoint problem for systems of conservation laws, CIMS Preprint.

    Google Scholar 

  101. R. LeVeque, A large time step generalization of Godunov's method for systems of conservation laws, SIAM J. Numer. Anal. 22(6) (1985), 1051–1073.

    CrossRef  MathSciNet  MATH  Google Scholar 

  102. R. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics, Birkhäuser, Basel 1992.

    CrossRef  MATH  Google Scholar 

  103. R. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems, Preprint.

    Google Scholar 

  104. D. Levermore and J.-G. Liu, Oscillations arising in numerical experiments, NATO ARW seies, Plenum, New-York (1993), To appear.

    MATH  Google Scholar 

  105. D. Levy and E. Tadmor, Non-oscillatory central schemes for the incompressible 2-D Euler equations, Math. Res. Lett., 4 (1997) 1–20.

    CrossRef  MathSciNet  MATH  Google Scholar 

  106. D. Li, Riemann problem for multi-dimensional hyperbolic conservation laws, Free boundary problems in fluid flow with applications (Montreal, PQ, 1990), 64–69, Pitman Res. Notes Math. Ser., 282.

    Google Scholar 

  107. C.-T. Lin and E. Tadmor, L1-stability and error estimates for approximate Hamiton-Jacobi solutions, in preparation.

    Google Scholar 

  108. P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pittman, London 1982.

    MATH  Google Scholar 

  109. P.L. Lisons, On kinetic equations, in Proc. Int'l Congress of Math., Kyoto, 1990, Vol. II, Math. Soc. Japan, Springer (1991), 1173–1185.

    Google Scholar 

  110. P. L. Lions, B. Perthame and P. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure and Appl. Math. 49 (1996), 599–638.

    CrossRef  MathSciNet  MATH  Google Scholar 

  111. P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of scalar conservation laws and related equations, J. Amer. Math. Soc. 7() (1994), 169–191

    CrossRef  MathSciNet  MATH  Google Scholar 

  112. P. L. Lions, B. Perthame and E. Tadmor, Kinetic formulation of the isentropic gas-dynamics equations and p-systems, Comm. Math. Phys. 163(2) (1994), 415–431.

    CrossRef  MathSciNet  MATH  Google Scholar 

  113. T.-P. Liu, The entropy condition and the admissibility of shocks, J. Math. Anal. Appl. 53 (1976), 78–88.

    CrossRef  MathSciNet  MATH  Google Scholar 

  114. T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys. 57 (1977), 135–148.

    CrossRef  MathSciNet  MATH  Google Scholar 

  115. B. Lucier, Error bounds for the methods of Glimm, Godunov and LeVeque, SIAM J. Numer. Anal. 22 (1985), 1074–1081.

    CrossRef  MathSciNet  MATH  Google Scholar 

  116. B. Lucier, Lecture Notes, 1993.

    Google Scholar 

  117. Y. Maday, S. M. OuldKaber and E. Tadmor, Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal. 30 (1993), 321–342.

    CrossRef  MathSciNet  MATH  Google Scholar 

  118. A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Springer-Verlag New-York, 1984.

    CrossRef  MATH  Google Scholar 

  119. A. Majda, J. McDonough and S. Osher, The Fourier method for nonsmooth initial data, Math. Comp. 30 (1978), 1041–1081.

    CrossRef  MathSciNet  MATH  Google Scholar 

  120. M.S. Mock, Systems of conservation laws of mixed type, J. Diff. Eq. 37 (1980), 70–88.

    CrossRef  MathSciNet  MATH  Google Scholar 

  121. M. S. Mock and P. D. Lax, The computation of discontinuous solutions of linear hyperbolic equations, Comm. Pure Appl. Math. 31 (1978), 423–430.

    CrossRef  MathSciNet  MATH  Google Scholar 

  122. K. W. Morton, Lagrange-Galerkin and characteristic-Galerkin methods and their applications, 3rd Int'l Conf. Hyperbolic Problems (B. Engquist &B. Gustafsson, eds.), Studentlitteratur, (1991), 742–755.

    Google Scholar 

  123. F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa 5 (1978), 489–507.

    MathSciNet  MATH  Google Scholar 

  124. F. Murat, A survey on compensated compactness, in ‘Contributions to Modern calculus of variations’ (L. Cesari, ed), Pitman Research Notes in Mathematics Series, John Wiley, New-York, 1987, 145–183.

    Google Scholar 

  125. R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Comm. Pure Appl. Math. 49 (1996), 1–30.

    CrossRef  MathSciNet  MATH  Google Scholar 

  126. H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comp. Phys. 87 (1990), 408–463.

    CrossRef  MathSciNet  MATH  Google Scholar 

  127. H. Nessyahu and E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer., Anal. 29 (1992), 1–15.

    CrossRef  MathSciNet  MATH  Google Scholar 

  128. S. Noelle, A note on entropy inequalities and error estimates for higher-order accurate finite volume schemes on irregular grids, Math. Comp. to appear.

    Google Scholar 

  129. O. A. Olěinik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. (2), 26 (1963), 95–172.

    CrossRef  MathSciNet  Google Scholar 

  130. S. Osher, Riemann solvers, the entropy condition, and difference approximations, SIAM J. Numer. Anal. 21 (1984), 217–235.

    CrossRef  MathSciNet  MATH  Google Scholar 

  131. S. Osher and F. Solomon, Upwind difference schemes for hyperbolic systems of conservation laws, Math. Comp. 38 (1982), 339–374.

    CrossRef  MathSciNet  MATH  Google Scholar 

  132. S. Osher and J. Sethian, Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys. 79 (1988), 12–49.

    CrossRef  MathSciNet  MATH  Google Scholar 

  133. S. Osher and E. Tadmor., On the convergence of difference approximations to scalar conservation laws, Math. of Comp. 50 (1988), 19–51.

    CrossRef  MathSciNet  MATH  Google Scholar 

  134. B. Perthame, Global existence of solutions to the BGK model of Boltzmann equations, J. Diff. Eq. 81 (1989), 191–205.

    CrossRef  MathSciNet  MATH  Google Scholar 

  135. B. Perthame, Second-order Boltzmann schemes for compressible Euler equations, SIAM J. Num. Anal. 29, (1992), 1–29.

    CrossRef  MathSciNet  MATH  Google Scholar 

  136. B. Perthame and E. Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys. 136 (1991), 501–517.

    CrossRef  MathSciNet  MATH  Google Scholar 

  137. K. H. Prendergast and K. Xu, Numerical hydrodynamics from gas-kinetic theory, J. Comput. Phys.. 109(1) (1993), 53–66.

    CrossRef  MathSciNet  MATH  Google Scholar 

  138. A. Rizzi and B. Engquist, Selected topics in the theory and practice of computational fluid dynamics, J. Comp. Phys. 72 (1987), 1–69.

    CrossRef  MathSciNet  MATH  Google Scholar 

  139. P. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43 (1981), 357–372.

    CrossRef  MathSciNet  MATH  Google Scholar 

  140. P. Roe, Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics, J. Comput. Phys. 63 (1986), 458–476.

    CrossRef  MathSciNet  MATH  Google Scholar 

  141. R. Richtmyer and K. W. Morton, Difference Methods for Initial-Value Problems, Interscience, 2nd ed., 1967.

    Google Scholar 

  142. F. Sabac, PhD. Thesis, Univ. S. Carolina, 1994.

    Google Scholar 

  143. R. Sanders, On Convergence of monotone finite difference schemes with variable spatial differencing, Math. of Comp. 40 (1983), 91–106.

    CrossRef  MathSciNet  MATH  Google Scholar 

  144. R. Sanders, A third-order accurate variation nonexpansive difference scheme for single conservation laws, Math. Comp. 51 (1988), 535–558.

    CrossRef  MathSciNet  MATH  Google Scholar 

  145. S. Schochet, The rate of convergence of spectral viscosity methods for periodic scalar conservation laws, SIAM J. Numer. Anal. 27 (1990), 1142–1159.

    CrossRef  MathSciNet  MATH  Google Scholar 

  146. S. Schochet, Glimm's scheme for systems with almost-planar interactions, Comm. Partial Differential Equations 16(8–9) (1991), 1423–1440.

    CrossRef  MathSciNet  MATH  Google Scholar 

  147. S. Schochet and E. Tadmor, Regularized Chapman-Enskog expansion for scalar conservation laws, Archive Rat. Mech. Anal. 119 (1992), 95–107.

    CrossRef  MathSciNet  MATH  Google Scholar 

  148. D. Serre, Richness and the classification of quasilinear hyperbolic systems, in “Multidimensional Hyperbolic Problems and Computations”, Minneapolis MN 1989, IMA Vol. Math. Appl. 29, Springer NY (1991), 315–333.

    Google Scholar 

  149. D. Serre, Systemés de Lois de Conservation, Diderot, Paris 1996.

    Google Scholar 

  150. D. Serre, private communication.

    Google Scholar 

  151. C. W. Shu, TVB uniformly high-order schemes for conservation laws, Math. Comp. 49 (1987) 105–121.

    CrossRef  MathSciNet  MATH  Google Scholar 

  152. C. W. Shu, Total-variation-diminishing time discretizations, SIAM J. Sci. Comput. 6 (1988), 1073–1084.

    CrossRef  MathSciNet  MATH  Google Scholar 

  153. C. W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comp. Phys. 77 (1988), 439–471.

    CrossRef  MathSciNet  MATH  Google Scholar 

  154. C. W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. II, J. Comp. Phys. 83 (1989), 32–78.

    CrossRef  MathSciNet  MATH  Google Scholar 

  155. W. Shyy, M.-H. Chen, R. Mittal and H.S. Udaykumar, On the suppression on numerical oscillations using a non-linear filter, J. Comput. Phys. 102 (1992), 49–62.

    CrossRef  MATH  Google Scholar 

  156. D. Sidilkover, Multidimensional upwinding: unfolding the mystery, Barriers and Challenges in CFD, ICASE workshop, ICASE, Aug, 1996.

    Google Scholar 

  157. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.

    CrossRef  MATH  Google Scholar 

  158. G. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, JCP 22 (1978) 1–31.

    MathSciNet  MATH  Google Scholar 

  159. G. Sod, Numerical Methods for Fluid Dynamics, Cambridge University Press, 1985.

    Google Scholar 

  160. A. SzepessyConvergence of a shock-capturing streamline diffusion finite element method for scalar conservation laws in two space dimensions, Math. of Comp. (1989), 527–545.

    Google Scholar 

  161. P. R. Sweby, High resolution schemes using flux limiters, for hyperbolic conservation laws SIAM J. Num. Anal. 21 (1984), 995–1011.

    CrossRef  MathSciNet  MATH  Google Scholar 

  162. E. Tadmor, The large-time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme Math. Comp. 43 (1984), no. 168, 353–368.

    CrossRef  MathSciNet  MATH  Google Scholar 

  163. E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp. 43 (1984), no. 168, 369–381.

    CrossRef  MathSciNet  MATH  Google Scholar 

  164. E. Tadmor, Convenient total variation diminishing conditions for nonlinear difference schemes, SIAM J. on Numer. Anal. 25 (1988), 1002–1014.

    CrossRef  MathSciNet  MATH  Google Scholar 

  165. E. Tadmor, Convergence of Spectral Methods for Nonlinear Conservation Laws, SIAM J. Numer. Anal. 26 (1989), 30–44.

    CrossRef  MathSciNet  MATH  Google Scholar 

  166. E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal. 28 (1991), 891–906.

    CrossRef  MathSciNet  MATH  Google Scholar 

  167. E. Tadmor, Super viscosity and spectral approximations of nonlinear conservation laws, in “Numerical Methods for Fluid Dynamics”, Proceedings of the 1992 Conference on Numerical Methods for Fluid Dynamics (M. J. Baines and K. W. Morton, eds.), Clarendon Press, Oxford, 1993, 69–82.

    Google Scholar 

  168. E. Tadmor, Approximate Solution of Nonlinear Conservation Laws and Related Equations, in “Recent Advances in Partial Differential Equations and Applications” Proceedings of the 1996 Venice Conference in honor of Peter D. Lax and Louis Nirenberg on their 70th Birthday (R. Spigler and S. Venakides eds.), AMS Proceedings of Symposia in Applied Mathematics 54, 1997, 321–368

    Google Scholar 

  169. E. Tadmor and T. Tassa, On the piecewise regularity of entropy solutions to scalar conservation laws, Com. PDEs 18 91993), 1631–1652.

    Google Scholar 

  170. T. Tang & Z. H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws, Math. Comp., 66 (1997), pp. 495–526.

    CrossRef  MathSciNet  MATH  Google Scholar 

  171. L. Tartar, Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics 39, Nonlinear Analysis and Mechanics, Heriott-Watt Symposium, Vol. 4 (R.J. Knopps, ed.) Pittman Press, (1975), 136–211.

    Google Scholar 

  172. L. Tartar, Discontinuities and oscillations, in Directions in PDEs, Math Res. Ctr Symposium (M.G. Crandall, P.H. Rabinowitz and R.E. Turner eds.) Academic Press (1987), 211–233.

    Google Scholar 

  173. T. Tassa, Applications of compensated compactness to scalar conservation laws, M.Sc. thesis (1987), Tel-Aviv University (in Hebrew).

    Google Scholar 

  174. B. Temple, No L1contractive metrics for systems of conservation laws, Trans. AMS 288(2) (1985), 471–480.

    MathSciNet  MATH  Google Scholar 

  175. Z.-H. TengParticle method and its convergence for scalar conservation laws, SIAM J. Num. Anal. 29 (1992) 1020–1042.

    CrossRef  MathSciNet  MATH  Google Scholar 

  176. Vasseur, Kinetic semi-discretization of scalar conservation laws and convergence using averaging lemmas, SIAM J. Numer. Anal.

    Google Scholar 

  177. A. I. Vol'pert, The spaces BV and quasilinear equations, Math. USSR-Sb. 2 (1967), 225–267.

    CrossRef  MathSciNet  MATH  Google Scholar 

  178. G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, 1974.

    Google Scholar 

  179. K. Yosida, Functional Analysis, Springer-Verlag, 1980.

    Google Scholar 

References

  1. P. Arminjon, D. Stanescu & M.-C. Viallon, A Two-Dimensional Finite Volume Extension of the Lax-Friedrichs and Nessyahu-Tadmor Schemes for Compressible Flow, (1995), preprint.

    Google Scholar 

  2. P. Arminjon, D. Stanescu & M.-C. Viallon, A two-dimensional finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for compressible flow, Preprint.

    Google Scholar 

  3. P. Arminjon & M.-C. Viallon, Généralisation du Schéma de Nessyahu-Tadmor pour Une Équation Hyperbolique à Deux Dimensions D'espace. C.R. Acad. Sci. Paris, t. 320, série I. (1995), pp. 85–88.

    MathSciNet  Google Scholar 

  4. F. Bereux & L. Sainsaulieu, A Roe-type Riemann Solver for Hyperbolic Systems with Relaxation Based on Time-Dependent Wave Decomposition, Numer. Math., 77, (1997), pp. 143–185.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. D. L. Brown & M. L. MinionPerformance of under-resolved two-dimensional incompressible flow simulations, J. Comp. Phys. 122, (1985) 165–183.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. P. Colella & P. Woodward, The piecewise parabolic method (PPM) for gas-dynamical simulations, JCP 54, 1984, pp. 174–201.

    MathSciNet  MATH  Google Scholar 

  7. B. Engquist & O. Runborg, Multi-phase computations in geometrical optics, J. Comp. Appl. Math., 1996, in press.

    Google Scholar 

  8. Erbes, A high-resolution Lax-Friedrichs scheme for Hyperbolic conservation laws with source term. Application to the Shallow Water equations. Preprint.

    Google Scholar 

  9. K. O. Friedrichs & P. D. Lax, Systems of Conservation Equations with a Convex Extension, Proc. Nat. Acad. Sci., 68, (1971), pp. 1686–1688.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. E. Godlewski & P.-A. Raviart, Hyperbolic Systems of Conservation Laws, Mathematics & Applications, Ellipses, Paris, 1991.

    MATH  Google Scholar 

  11. 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.

    MathSciNet  MATH  Google Scholar 

  12. A. Harten, High Resolution Schemes for Hyperbolic Conservation Laws, JCP, 49, (1983), pp. 357–393.

    MathSciNet  MATH  Google Scholar 

  13. A. Harten, B. Engquist, S. Osher & S. R. Chakravarthy, Uniformly high order accurate essentially non-oscillatory schemes. III, JCP 71, 1982, pp. 231–303.

    MathSciNet  MATH  Google Scholar 

  14. H. T. Huynh, A piecewise-parabolic dual-mesh method for the Euler equations, AIAA-95-1739-CP, The 12th AIAA CFD Conf., 1995.

    Google Scholar 

  15. G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher & E. Tadmor, High-resolution Non-Oscillatory Central Schemes with Non-Staggered Grids for Hyperbolic Conservation Laws, SIAM Journal on Num. Anal., to appear.

    Google Scholar 

  16. G.-S. Jiang & E. Tadmor, Nonoscillatory Central Schemes for Multidimensional Hyperbolic Conservation Laws, SIAM J. Scie. Comp., to appear.

    Google Scholar 

  17. S. Jin, private communication.

    Google Scholar 

  18. S. Jin and Z. Xin, The relaxing schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995) 235–277.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. B. van Leer, Towards the Ultimate Conservative Difference Scheme, V. A. Second-Order Sequel to Godunov's Method, JCP, 32, (1979), pp. 101–136.

    Google Scholar 

  20. R. Kupferman, Simulation of viscoelastic fluids: Couette-Taylor flow, J. Comp. Phys., to appear.

    Google Scholar 

  21. R. Kupferman, A numerical study of the axisymmetric Couette-Taylor problem using a fast high-resolution second-order central scheme, SIAM. J. Sci. Comp., to appear.

    Google Scholar 

  22. R. Kupferman & E. Tadmor, A Fast High-Resolution Second-Order Central Scheme for Incompressible Flow s, Proc. Nat. Acad. Sci. 94 (1997), 4848–4852

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. R. J. LeVeque, Numerical Methods for Conservation Laws, Lectures in Mathematics, Birkhauser Verlag, Basel, 1992.

    CrossRef  MATH  Google Scholar 

  24. D. Levy, Third-order 2D Central Schemes for Hyperbolic Conservation Laws, in preparation.

    Google Scholar 

  25. D. Levy & E. Tadmor, Non-oscillatory Central Schemes for the Incompressible 2-D Euler Equations, Math. Res. Let., 4, (1997), pp. 321–340.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. D. Levy & E. Tadmor, Non-oscillatory boundary treatment for staggered central schemes, preprint.

    Google Scholar 

  27. X.-D. Liu & P. D. Lax, Positive Schemes for Solving Multi-dimensional Hyperbolic Systems of Conservation Laws, Courant Mathematics and Computing Laboratory Report, Comm. Pure Appl. Math.

    Google Scholar 

  28. X.-D. Liu & S. Osher, Nonoscillatory High Order Accurate Self-Similar Maximum Principle Satisfying Shock Capturing Schemes I, SINUM, 33, no. 2 (1996), pp. 760–779.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. X.-D. Liu & E. Tadmor, Third Order Nonoscillatory Central Scheme for Hyperbolic Conservation Laws, Numer. Math., to appear.

    Google Scholar 

  30. H. Nessyahu, Non-oscillatory second order central type schemes for systems of nonlinear hyperbolic conservation laws, M.Sc. Thesis, Tel-Aviv University, 1987.

    Google Scholar 

  31. H. Nessyahu & E. Tadmor, Non-oscillatory Central Differencing for Hyperbolic Conservation Laws, JCP, 87, no. 2 (1990), pp. 408–463.

    MathSciNet  MATH  Google Scholar 

  32. H. Nessyahu, E. Tadmor & T. Tassa, On the convergence rate of Godunov-type schemes, SINUM 31, 1994, pp. 1–16.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. S. Osher & E. Tadmor, On the Convergence of Difference Approximations to Scalar Conservation Laws, Math. Comp., 50, no. 181 (1988), pp. 19–51.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. P. L. Roe, Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes, JCP, 43, (1981), pp. 357–372.

    MathSciNet  MATH  Google Scholar 

  35. A. Rogerson & E. Meiburg, A numerical study of the convergence properties of ENO schemes, J. Sci. Comput., 5, 1990, pp. 127–149.

    CrossRef  MATH  Google Scholar 

  36. O. Runborg, Multiphase Computations in Geometrical Optics, UCLA CAM report no. 96-52 (1996).

    Google Scholar 

  37. V. Romano & G. Russo, Numerical solution for hydrodynamical models of semiconductors, IEEE, to appear.

    Google Scholar 

  38. A. M. Anile, V. Romano & G. Russo, Extended hydrodymnamical model of carrier transport in semiconductors, Phys. Rev. B., to appear.

    Google Scholar 

  39. F. Bianco, G. Puppo & G. Russo, High order central schemes for hyperbolic systems of conservation laws, SIAM J. Sci. Comp., to appear.

    Google Scholar 

  40. R. Sanders, A Third-order Accurate Variation Nonexpansive Difference Scheme for Single Conservation Laws, Math. Comp., 41 (1988), pp. 535–558.

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. R. Sanders R. & A. Weiser, A High Resolution Staggered Mesh Approach for Nonlinear Hyperbolic Systems of Conservation Laws, JCP, 1010 (1992), pp. 314–329.

    MathSciNet  MATH  Google Scholar 

  42. P. K. Sweby, High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws, SINUM, 21, no. 5 (1984), pp. 995–1011.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. C.-W. Shu, Numerical experiments on the accuracy of ENO and modified ENO schemes, JCP 5, 1990, pp. 127–149.

    MATH  Google Scholar 

  44. G. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, JCP 22, 1978, pp. 1–31.

    MathSciNet  MATH  Google Scholar 

  45. E. Tadmor & C. C. Wu, Central Scheme for the Multidimensional MHD Equations, in preparation.

    Google Scholar 

  46. P. Woodward & P. Colella, The numerical simulation of two-dimensional fluid flow with strong shocks, JCP 54, 1988, pp. 115–173.

    MathSciNet  MATH  Google Scholar 

References

  1. S. Abarbanel, D. Gottlieb & E. Tadmor, Spectral methods for discontinuous problems, in “Numerical Analysis for Fluid Dynamics II” (K.W. Morton and M.J. Baines, eds.), Oxford University Press, 1986, pp. 129–153.

    Google Scholar 

  2. Ø. Andreassen, I. Lie & C.-E. Wassberg, The spectral viscosity method applied to simulation of waves in a stratified atmosphere, J. Comp. Phys. 110 (1994) pp. 257–273.

    CrossRef  MATH  Google Scholar 

  3. W. Cai, D. Gottlieb & A. Harten, Cell averaging Chebyshev method for hyperbolic problems, Compu. and Math. with Appl., 24 (1992).

    Google Scholar 

  4. C. Canuto, M.Y. Hussaini, A. Quarteroni & T. Zang, Spectral Methods with Applications to Fluid Dynamics, Springer-Verlag, 1987.

    Google Scholar 

  5. G.-Q. Chen, Q. Du & E. Tadmor, Spectral viscosity approximations to multidimensional scalar conservation laws, Math. of Comp. 61 (1993), 629–643.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. R. DiPerna, Convergence of approximate solutions to systems of conservation laws, Arch. Rat. Mech. Anal., Vol. 82, pp. 27–70 (1983).

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. D.C. Fritts, T. L. Palmer, Ø. Andreassen, and I. LieEvolution and breakdown of Kelvin-Helmholtz billows in stratified compressible flows. 1. Comparison of two-and three-dimensional flows, J. Atmos. Sci. 53 (1996) pp. 3173–3191.

    CrossRef  Google Scholar 

  8. D. Gottlieb, Private communication.

    Google Scholar 

  9. D. Gottlieb, C.-W. Shu & H. Vandeven, Spectral reconstruction of a discontinuous periodic function, submitted to C. R. Acad. Sci. Paris.

    Google Scholar 

  10. D. Gottlieb & E. Tadmor, Recovering pointwise values of discontinuous data with spectral accuracy, in “Progress and Supercomputing in Computational Fluid Dynamics” (E. M. Murman and S.S. Abarbanel eds.), Progress in Scientific Computing, Vol. 6, Birkhauser, Boston, 1985, pp. 357–375.

    CrossRef  Google Scholar 

  11. H.-O. Kreiss & J. Oliger, Stability of the Fourier method, SINUM 16 (1979, pp. 421–433.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. H. Ma, Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws, SINUM, to appear.

    Google Scholar 

  13. H. Ma, Chebyshev-Legendre spectral super viscosity method for nonlinear conservation laws, SINUM, to appear.

    Google Scholar 

  14. B. van Leer, Toward the ultimate conservative difference schemes. V. A second order sequel to Godunov method, J. Compt. Phys. 32, (1979), pp. 101–136.

    CrossRef  Google Scholar 

  15. I. Lie, On the spectral viscosity method in multidomain Chebyshev discretizations, preprint.

    Google Scholar 

  16. F. Murat, Compacité per compensation, Ann. Scuola Norm. Sup. Disa Sci. Math. 5 (1978), pp. 489–507 and 8 (1981), pp. 69–102.

    MathSciNet  MATH  Google Scholar 

  17. Y. Maday & E. Tadmor, Analysis of the spectral viscosity method for periodic conservation laws, SINUM 26, 1989, pp. 854–870.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. Y. Maday, S.M. Ould Kaber & E. Tadmor, Legendre pseudospectral viscosity method for nonlinear conservation laws, SINUM 30, 1993, pp. 321–342.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. A. Majda, J. McDonough & S. Osher, The Fourier method for nonsmooth initial data, Math. Comp. 30, 1978, pp. 1041–1081.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. S.M. Ould Kaber, Filtrage d'ordre infini en non périodique, in Thèse de Doctorat, Université Pierre et Marie Curie, Paris, 1991.

    Google Scholar 

  21. S. Schochet, The rate of convergence of spectral viscosity methods for periodic scalar conservation laws, SINUM 27, 1990, pp. 1142–1159.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. C.-W. Shu & S. Osher, Efficient Implementation of Essentially Nonoscillatory Shock-Capturing schemes, II, J. Comp. Phys., 83 (1989), pp. 32–78.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. C.-W. Shu & P. Wong, A note on the accuracy of spectral method applied to nonlinear conservation laws, J. Sci. Comput., v10 (1995), pp. 357–369.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. A. Szepessy, Measure valued solutions to scalar conservation laws with boundary conditions, Arch. Rat. Mech. 107, 181–193 (1989).

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. E. Tadmor, The exponential accuracy of Fourier and Chebyshev differencing methods SINUM 23, 1986, pp. 1–10.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SINUM 26, 1989, pp. 30–44.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. E. Tadmor, Semi-discrete approximations to nonlinear systems of conservation laws; consistency and stability imply convergence, ICASE Report no. 88-41.

    Google Scholar 

  28. E. Tadmor, Shock capturing by the spectral viscosity method., Computer Methods in Appl. Mech. Engineer. 80 1990, pp. 197–208.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. E. Tadmor, Total-variation and error estimates for spectral viscosity approximations, Math. Comp. 60, 1993, pp. 245–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. E. Tadmor, Super viscosity and spectral approximations of nonlinear conservation laws, in “Numerical Methods for Fluid Dynamics IV”, Proceedings of the 1992 Conference on Numerical Methods for Fluid Dynamics, (M. J. Baines and K. W. Morton, eds.), Clarendon Press, Oxford, 1993, pp. 69–82.

    Google Scholar 

  31. L. Tartar, Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics 39, Nonlinear Analysis and Mechanics, Heriott-Watt Symposium, Vol. 4 (R.J. Knopps, ed.) Pittman Press, pp. 136–211 (1975).

    Google Scholar 

  32. H. Vandeven, A family of spectral filters for discontinuous problems, J. Scientific Comput. 8, 1991, pp. 159–192.

    CrossRef  MathSciNet  MATH  Google Scholar 

References

  1. G. Barles & P.E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asympt. Anal. 4 (1991), 271–283.

    MathSciNet  MATH  Google Scholar 

  2. F. Bouchut & B. Perthame, Kruzkov's estimates for scalar conservation laws revisited, Universite D'Orleans, preprint, 1996.

    Google Scholar 

  3. Y. Brenier, Roe's scheme and entropy solution for convex scalar conservation laws, INRIA Report 423, France 1985.

    Google Scholar 

  4. Y. Brenier & S. Osher, The discrete one-sided Lipschitz condition for convex scalar conservation laws. 1988, SIAM J. of Num. Anal. Vol. 25, 1, pp. 8–23.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. A. J. Chorin, Random choice solution of hyperbolic systems, J. Comp. Phys., 22 (1976) pp. 517–533.

    CrossRef  MathSciNet  MATH  Google Scholar 

  6. I. L. Chern, Stability theorem and truncation error analysis for the Glimm scheme and for a front tracking method for flows with strong discontinuities, Comm. Pure Appl. Math., XLII (1989), pp. 815–844.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. B. Cockburn, F. Coquel & P. LeFloch, Convergence of finite volume methods for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), 687–705.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. M. G. Crandall & P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc. 277 (1983), 1–42.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. M. G. Crandall & A. Majda, Monotone difference approximations for scalar conservation laws, Math. Comp., 34 (1980), 1–21.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), pp. 697–715.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. J. B. Goodman & R. J. LeVeque, A geometric approach to high resolution TVD schemes, SIAM J. Numer. Anal., 25 (1988), pp. 268–284.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. D. Gottlieb & E. Tadmor, Recovering Pointwise Values of Discontinuous Data within Spectral Accuracy, in “Progress and supercomputing in Computational Fluid Dyamics”, Progress in Scientific Computing, Vol. 6 (E. M. Murman and S. S. Abarbanel, eds.), Birkhauser, Boston, 1985, 357–375.

    CrossRef  Google Scholar 

  13. J. Goodman & P. D. Lax, On dispersive difference schemes. I, Comm. Pure Appl. Math. 41 (1988), 591–613.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. D. Hoff & J. Smoller, Error bounds for the Glimm scheme for a scalar conservation law, Trans. Amer. Math. Soc., 289 (1988), pp. 611–642.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. N.N. Kuznetsov, On stable methods for solving nonlinear first order partial differntial equations in the class of discontinuous solutions, Topics in Num. Anal. III, Proc. Royal Irish Acad. Conf. Trinity College, Dublin (1976), pp. 183–192.

    Google Scholar 

  16. S. N. Kružkov, The method of finite difference for a first order non-linear equation with many independent variables, USSR Comput Math. and Math. Phys. 6 (1966), 136–151. (English Trans.)

    CrossRef  Google Scholar 

  17. S. N. Krushkov, First-order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217–243.

    CrossRef  Google Scholar 

  18. A. Kurganov & E. Tadmor, Stiff systems of hyperbolic conservation laws. Convergence and error estimates, SIMA, in press.

    Google Scholar 

  19. C.-T. Lin & E. TadmorL1-Stability and error estimates for approximate Hamilton-Jacobi solutions, preprint.

    Google Scholar 

  20. P. L. Lions, Generalized Solutions of Hamilton-Jacobi Equations, Pittman, London 1982.

    MATH  Google Scholar 

  21. T. P. Liu, The deterministic version of the Glimm scheme, Comm. Math. Phys., 57 (1977), pp. 135–148.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. B. Lucier, Error bounds for the methods of Glimm, Godunov, and LeVeque, SIAM J. of Numer. Anal., 22 (1985), pp. 1074–1081.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. Y. Maday & E. Tadmor, Analysis of the spectral viscosity method for periodic conservation laws, SINUM 26, 1989, pp. 854–870.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. H. Nessyahu & E. Tadmor, The convergence rate of approximate solutions for nonlinear scalar conservation laws, SIAM J. Numer. Anal. 29 (1992), 1–15.

    CrossRef  MathSciNet  MATH  Google Scholar 

  25. H. Nessyahu & T. Tassa, Convergence rates of approximate solutions to conmservation laws with initial rarefactions, SIAM J. Numer. Anal., 31 (1994), 628–654.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. H. Nessyahu, E. Tadmor & T. Tassa, The convergence rate of Godunov type schemes, SIAM J. Numer. Anal. 31 (1994), 1–16.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. H. Nessyahu, Convergence rate of approximate solutions to weakly coupled nonlinear systems, Math. Comp. 65 (1996) pp. 575–586.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion, Phys. Rev. A, 40(1989), 7193–6.

    CrossRef  MathSciNet  Google Scholar 

  29. R. Richtmyer & K. W. Morton, Difference methods for initial-value problems, 2nd ed., Interscience, New York, 1967.

    MATH  Google Scholar 

  30. S. Schochet, The rate of convergence of spectral viscosity methods for periodic scalar conservation laws, SINUM 27, 1990, pp. 1142–1159.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. R. Sanders, On convergence of monotone finite difference schemes with variable spatial differencing, Math. of Comp., 40 (1983), pp. 91–106.

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. J. Smoller Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.

    CrossRef  MATH  Google Scholar 

  33. S. Schochet & E. Tadmor, The regularized Chapman-Enskog expansion for scalar conservation laws, Arch. Rational Mech. Anal., 119 (1992), pp. 95–107.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. E. Tadmor, The large time behavior of the scalar, genuinely nonlinear Lax-Friedrichs scheme, Math. of Comp., 43, 168 (1984), pp. 353–368.

    CrossRef  MathSciNet  MATH  Google Scholar 

  35. E. Tadmor, Numerical viscosity and the entropy condition for conservative difference schemes, Math. Comp., 43 (1984), pp. 369–381.

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. E. Tadmor, The numerical viscosity of entropy stable schemes for systems of conservation laws. I, Math. of Comp., 49 (1987), pp. 91–103.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. E. Tadmor, Semi-discrete approximations to nonlinear systems of conservation laws; consistency and L-stability imply convergence, ICASE ICASE Report No. 88-41.

    Google Scholar 

  38. E. Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal., 26 (1989), pp. 30–44.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. E. Tadmor, Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal., 28 (1991), pp. 811–906.

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. E. Tadmor, Total variation and error estimates fo spectral viscosity approximations, Math. Comp., 60 (1993), pp. 245–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. E. Tadmor & T. Tang, The pointwise convergence rate for piecewise smooth solutions for scalar conservatin laws, in preparation.

    Google Scholar 

  42. T. Tang & Z. H. Teng, Viscosity methods for piecewise smooth solutions to scalar conservation laws, Math. Comp., 66 (1997), pp. 495–526.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. T. Tang & P.-W. Zhang, Optimal L1-rate of convergence for viscosity method and monotone schemes to piecewise constant solutions with shocks, SIAM J. Numer. Anala. 34 (1997), pp. 959–978.

    CrossRef  Google Scholar 

References

  1. C. Bardos, F. Golse & D. Levermore, Fluid dynamic limits of kinetic equations II: convergence proofs of the Boltzmann equations, Comm. Pure Appl. Math. XLVI (1993), 667–754.

    CrossRef  MathSciNet  MATH  Google Scholar 

  2. Y. Brenier, Résolution d'équations d'évolution quasilinéaires en dimension N d'espace à l'aide d'équations linéaires en dimension N+1, J. Diff. Eq. 50 (1983), 375–390.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. G.-Q. Chen, Q. Du & E. Tadmor, Spectral viscosity approximation to multidimensional scalar conservation laws, Math. of Comp. 57 (1993).

    Google Scholar 

  4. B. Cockburn, F. Coquel & P. LeFloch, Convergence of finite volume methods for multidimensional conservation laws, SIAM J. Numer. Anal. 32 (1995), 687–705.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. C. Cercignani, The Boltzmann Equation and its Applications, Appl. Mathematical Sci. 67, Springer, New-York, 1988.

    MATH  Google Scholar 

  6. G.-Q. Chen, The theory of compensated compactness and the system of isentropic gas dynamics, Preprint MCS-P154-0590, Univ. of Chicago, 1990.

    Google Scholar 

  7. R. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91 (1983), 1–30.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. R. DiPerna, Measure-valued solutions to conservation laws, Arch. Rat. Mech. Anal. 88 (1985), 223–270.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. R. DiPerna & P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math. 130 (1989), 321–366.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. R. DiPerna & P. L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 42 (1989), 729–757.

    CrossRef  MathSciNet  MATH  Google Scholar 

  11. R. DiPerna, P. L. Lions & Y. Meyer, Lpregularity of velocity averages, Ann. I.H.P. Anal. Non Lin. 8(3–4) (1991), 271–287.

    MathSciNet  MATH  Google Scholar 

  12. P. Gérard, Microlocal defect measures, Comm. PDE 16 (1991), 1761–1794.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. F. Golse, P. L. Lions, B. Perthame & R. Sentis, Regularity of the moments of the solution of a transport equation, J. of Funct. Anal. 76 (1988), 110–125.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. Y. Giga & T. Miyakawa, A kinetic construction of global solutions of firstorder quasilinear equations, Duke Math. J. 50 (1983), 505–515.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. C. Johnson & A. Szepessy, Convergence of a finite element methods for a nonlinear hyperbolic conservation law, Math. of Comp. 49 (1988), 427–444.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. C. Johnson, A. Szepessy & P. Hansbo, On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. of Comp. 54 (1990), 107–129.

    CrossRef  MathSciNet  MATH  Google Scholar 

  17. Y. Kobayashi, An operator theoretic method for solving ut=Δϕ(u), Hiroshima Math. J. 17 (1987) 79–89.

    MathSciNet  Google Scholar 

  18. D. Kröner, S. Noelle & M. Rokyta, Convergence of higher order upwind finite volume schemes on unstructured grids for scalar conservation laws in several space dimensions, Numer. Math. 71 (1995) 527–560.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. D. Kröner & M. Rokyta, Convergence of Upwind Finite Volume Schemes for Scalar Conservation Laws in two space dimensions, SINUM 31 (1994) 324–343.

    CrossRef  MATH  Google Scholar 

  20. P. D. Lax, Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves (SIAM, Philadelphia, 1973).

    CrossRef  MATH  Google Scholar 

  21. P. L. Lions, B. Perthame & P. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure and Appl. Math. 49 (1996), 599–638.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. P. L. Lions, B. Perthame & E. Tadmor, Kinetic formulation of scalar conservation laws and related equations, J. Amer. Math. Soc. 7(1) (1994), 169–191.

    CrossRef  MathSciNet  MATH  Google Scholar 

  23. P. L. Lions, B. Perthame & E. Tadmor, Kinetic formulation of the isentropic gas-dynamics equations and p-systems, Comm. Math. Phys. 163(2) (1994), 415–431.

    CrossRef  MathSciNet  MATH  Google Scholar 

  24. S. Noelle & M. WestdickenbergConvergence of finite volume schemes. A new convergence proof for finite volume schemes using the kinetic formulation of conservation laws, Preprint.

    Google Scholar 

  25. O. A. OlěinikDiscontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. (2), 26 (1963), 95–172.

    CrossRef  MathSciNet  Google Scholar 

  26. B. Perthame, Global existence of solutions to the BGK model of Boltzmann equations, J. Diff. Eq. 81 (1989), 191–205.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. B. Perthame, Second-order Boltzmann schemes for compressible Euler equations, SIAM J. Num. Anal. 29, (1992), 1–29.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. B. Perthame & E. Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys. 136 (1991), 501–517.

    CrossRef  MathSciNet  MATH  Google Scholar 

  29. K. H. Prendergast & K. Xu, Numerical hydrodynamics from gas-kinetic theory, J. Comput. Phys. 109(1) (1993), 53–66.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.

    CrossRef  MATH  Google Scholar 

  31. L. Tartar, Compensated compactness and applications to partial differential equations, in Research Notes in Mathematics 39, Nonlinear Analysis and Mechanics, Heriott-Watt Symposium, Vol. 4 (R.J. Knopps, ed.) Pittman Press, (1975), 136–211.

    Google Scholar 

  32. L. Tartar, Discontinuities and oscillations, in Directions in PDEs, Math Res. Ctr Symposium (M.G. Crandall, P.H. Rabinowitz and R.E. Turner eds.) Academic Press (1987), 211–233.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 1998 Springer-Verlag

About this chapter

Cite this chapter

Tadmor, E. (1998). Approximate solutions of nonlinear conservation laws. In: Quarteroni, A. (eds) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol 1697. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0096352

Download citation

  • DOI: https://doi.org/10.1007/BFb0096352

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64977-9

  • Online ISBN: 978-3-540-49804-9

  • eBook Packages: Springer Book Archive