CONVERGENCE RATE ESTIMATES

A collection of selected references on

Piecewise regularity and convergence rate estimates
for nonlinear transport equations




  • D. Hoff and J. Smoller
    Error bounds for finite-difference approximations for a class of nonlinear parabolic systems
    Mathematics of Computation 45 (1985) 35-49.

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  • S. Schochet
    The rate of convergence of spectral-viscosity methods for periodic scalar conservation laws
    SIAM Journal of Numerical Analysis 27 (1990) 1142-1159.

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  • E. Tadmor
    Essentially non-oscillatory spectral viscosity approximations
    in "Hyperbolic Problems - Theory, Numerical Methods and Applications", Proceedings of the 3rd International Conference on Hyperbolic Problems, Vol. II (B. Engquist and B. Gustafsson, eds.), Studentlitteratur and Chartwell-Bratt (1991) 861-873..

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  • E. Tadmor
    Local error estimates for discontinuous solutions of nonlinear hyperbolic equations
    SIAM Journal of Numerical Analysis 28 (1991) 891-906.

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  • H. Nessyahu and E. Tadmor
    The convergence rate of approximate solutions for nonlinear scalar conservation laws
    SIAM Journal of Numerical Analysis 29 (1991) 1505-1519.

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  • E. Tadmor and T. Tassa
    On the piecewise smoothness of entropy solutions to scalar conservation laws
    Communications on Partial Differential Equations 18 (1993) 1631-1652.

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  • E. Tadmor
    Total variation and error estimates for spectral viscosity approximations
    Mathematics of Computation 60 (1993) 245-256.

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  • B. Cockburn, F. Coquel and P. LeFloch
    An error estimate for finite volume methods for multidimensional conservation laws
    Mathematics of Computations 63 (1994) 77-103.

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  • H. Nessyahu, E. Tadmor and T. Tassa
    The convergence rate of Godunov type schemes
    SIAM Journal of Numerical Analysis 31 (1994) 1-16.

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  • H. Nessyahu and T. Tassa
    Convergence rate of approximate solutions to conservation laws with initial rarefacrtions
    SIAM Journal of Numerical Analysis 31 (1994) 628-654.

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  • H. Nessyahu
    Convergence rate of approximate solutions to weakly coupled nonlinear systems
    Mathematics of Computation 65 (1996) 575-585-342.

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  • F. Bouchut, Ch. Bourdarias and B. Perthame
    A MUSCL method satisfying all the numerical entropy inequalities
    Mathematics of Computation 65 (1996) 65 (1996), 1439-1461.

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  • A. Kurganov and E. Tadmor
    Stiff systems of hyperbolic conservation laws: convergence and error estimates
    SIAM Journal of Numerical Analysis 28 (1997) 1446-1456.

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  • Florin Sabac
    The optimal convergence rate of monotone finite difference methods for hyperbolic conservation laws
    SIAM J Numerical Analysis 34(6) (1997) 2306-2318.

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  • F. James
    Convergence results for some conservation laws with a reflux boundary condition and a relaxation term arising in chemical engineering
    SIAM Journal of Numerical Analysis 29 (1998) 1200-1223.

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  • W. -C. Wang
    On L1 convergence rate of viscous and numerical approximate solutions of genuinely nonlinear scalar conservation laws
    SIAM Journal of Numerical Analysis 30 (1998) 38-52.

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  • E. Tadmor and T. Tang
    Pointwise convergence rate for nonlinear conservation laws
    in "Hyperbolic Problems:Theory, Numerics, Applications", Proceedings of the 7th int'l Conference in 1998, Vol. ISNM v. 130 (M. Fey and R. Jeltsch, eds.), Birkhauser, Zurich (1999) 925-934..

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  • W. Shen, A. Tveito and R. Winther
    On the zero relaxation limit for a system modeling the motion of viscoelastic solid
    SIAM Journal of Numerical Analysis 30 (1999) 1115-1135.

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  • E. Tadmor and T. Tang
    Pointwise error estimates for scalar conservation laws with piecewise smooth solutions
    SIAM Journal of Numerical Analysis 36 (1999) 1739-1758.

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  • T. Tang and J. Wang
    Convergence of MUSCL relaxing schemes to the relaxed schemes of conservation laws with stiff source terms
    Journal of Scientific Computing 15 (2000) 173-195.

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  • T. Tang
    Error estimates for approximate solutions for nonlinear scalar conservation laws
    in "Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 8th international conference, Vol. v. 141 (H. Freistuhler and G. Warnecke, eds.), Birkhauser, Magdeburg (2000) 873-882..

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  • E. Tadmor and T. Tang
    Pointwise error estimates for relaxation approximations to conservation laws
    SIAM Journal of Numerical Analysis 32 (2000) 870-886.

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  • C.-T. Lin and E. Tadmor
    High-resolution non oscillatory central schemes for Hamilton-Jacobi equations
    SIAM Journal of Numerical Analysis 21 (2000) 2163-2186.

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  • L. Gosse and C. Makridakis
    Two a posteriori error estimates for one-dimensional scalar conservation laws
    SIAM Journal of Numerical Analysis 38 (2000) 964-988.

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  • G. Petrova and B. Popov
    Linear transport equations with discontinuous coefficients
    (2001).

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  • H. Berestycki, S. Kamin and G. Sivashinsky
    Metastability in a flame front evolution equation
    Interfaces and Free Boundaries 3 (2001) 361-392.

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  • H. Liu, J. Wang and G. Warnecke
    Convergence of a splitting scheme applied to the Ruijgrok-Wu model of the Boltzmann equation
    Journal of Computational and Applied Mathematics 134 (2001) 343-367.

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  • Y.-X. Kan, T.Tang and Z.-H. Teng
    On the piecewise smooth solutions to non-homogeneous scalar conservation laws
    Journal of Differential Equations 175 (2001) 27-50.

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  • C. T. Lin and E. Tadmor
    L1-stability and error estimates for approximate Hamilton-Jacobi solutions
    Numerische Mathematik 87 (2001) 701-735.

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  • C. Chainais-Hillairet and S. Champier
    Finite volume schemes for non-homogeneous scalar conservation laws: error estimates
    Numerische Mathematik 93 (2001) 607-639.

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  • H. Liu, J. Wang and G. Warnecke
    The Lip+ stability and error estimates for a relaxation scheme
    SIAM Journal of Numerical Analysis 38 (2001) 1154-1170.

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  • T. Tang and Z.-H. Teng
    On the regularity of approximate solutions to conservation laws with piecewise smooth solutions
    SIAM Journal of Numerical Analysis 38 (2001) 1483-1495.

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  • A, J, Briggs, J. R. Claisse and C. M. Elliott
    Finite-difference approximation of a one-dimensional Hamilton-Jacobi system arising in superconductivity
    IMA Journal of Numerocal Analysis 22 (2002) 89-131.

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  • S. Karni, A. Kurganov and G. Petrova
    Smoothness indicators for adaptive algorithms
    Journal of Computational Physics 178 (2002) 323-341.

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  • L. Gosse and F. James
    Convergence results for an inhomogeneous system arising in various high frequency approximations
    Numerische Mathematik 90 (2002) 721-753.

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  • M. Kuther
    A priori error estimates for approximate solutions to convex conservation laws
    Numerische Mathematik 93 (2003) 697-728.

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  • J.-H. Wang and J.-H. Wang
    Existence and decay rate of solutions to the generalized Burgers equations
    Journal of Mathematical Analysis and Applications 284 (2003) 213-235.

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  • T. Tang, Z.-H. Teng and Z.-P. Xin
    Fractional rate of convergence for viscous approximation to non-convex conservation laws
    SIAM J Math Analysis 35(1) (2003) 98-122.

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  • H. X. Liu and T. Pan
    Contsruction of solutions and L1-error estimates of viscous methods for scalar conservation laws with boundaries
    Acta Matemthica Sinica DOI 10.1007/S10114-005-0638-x (2005).

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  • S. Karni and A. Kurganov
    Local error analysis for approximate solutions of hyperbolic conservation laws
    Advances in Computational Mathematics 22 (2005) 79-99.

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  • M. Giles and S. Ulbrich
    Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 1: Linearized approximations and linearized output functionals
    SIAM Journal on Numerical Analysis 48 (3) (2010) 882-904.

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  • Jochen Sch├╝tz, Sebastian Noelle, Christina Steiner and Georg May
    A note on sdjoint error estimation for one-dimensional stationary balance laws with shocks
    SIAM J. Numerical Analysis 51(1) (2013) 126-136.

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