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Volumn 35, Issue 1, 2001, Pages 17-33

Adaptive finite element relaxation schemes for hyperbolic conservation laws

Author keywords

Adaptive methods; Conservation laws; Finite elements

Indexed keywords


EID: 0035631409     PISSN: 0764583X     EISSN: None     Source Type: Journal    
DOI: 10.1051/m2an:2001105     Document Type: Article
Times cited : (17)

References (51)
  • 1
    • 85065353078 scopus 로고    scopus 로고
    • Convergence of relaxation schemes for conservation laws
    • D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws. Appl. Anal. 61 (1996) 163-193.
    • (1996) Appl. Anal. , vol.61 , pp. 163-193
    • Aregba-Driollet, D.1    Natalini, R.2
  • 2
    • 0041687749 scopus 로고    scopus 로고
    • Discrete kinetic schemes for multidimensional systems of conservation laws
    • D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973-2004.
    • (2000) SIAM J. Numer. Anal. , vol.37 , pp. 1973-2004
    • Aregba-Driollet, D.1    Natalini, R.2
  • 3
    • 84864179749 scopus 로고    scopus 로고
    • The adaptive finite element method
    • University of Texas at Austin
    • I. Babuška, The adaptive finite element method. TICAM Forum Notes no 7, University of Texas at Austin (1997).
    • (1997) TICAM Forum Notes No 7
    • Babuška, I.1
  • 4
    • 0022047088 scopus 로고
    • Basic principles of feedback and adaptive approaches in the finite element method
    • I. Babuška and W. Gui, Basic principles of feedback and adaptive approaches in the finite element method. Comput. Methods Appl. Mech. Engrg. 55 (1986) 27-42.
    • (1986) Comput. Methods Appl. Mech. Engrg. , vol.55 , pp. 27-42
    • Babuška, I.1    Gui, W.2
  • 5
    • 0001762370 scopus 로고    scopus 로고
    • Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems
    • M. Berger and R. LeVeque, Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35 (1998) 2298-2316.
    • (1998) SIAM J. Numer. Anal. , vol.35 , pp. 2298-2316
    • Berger, M.1    LeVeque, R.2
  • 6
    • 0033246851 scopus 로고    scopus 로고
    • Construction of BGK models with a family of kinetic entropies for a given system of conservation laws
    • F. Bouchut, Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999) 113-170.
    • (1999) J. Statist. Phys. , vol.95 , pp. 113-170
    • Bouchut, F.1
  • 8
    • 84980139661 scopus 로고
    • The fluid dynamical limit of a nonlinear model Boltzmann equation
    • R.E. Caflisch and G.C. Papanicolaou, The fluid dynamical limit of a nonlinear model Boltzmann equation. Comm. Pure Appl. Math. 32 (1979) 589-616.
    • (1979) Comm. Pure Appl. Math. , vol.32 , pp. 589-616
    • Caflisch, R.E.1    Papanicolaou, G.C.2
  • 9
    • 84990689276 scopus 로고
    • Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm
    • G.-Q. Chen, C.D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994) 789-830.
    • (1994) Pure Appl. Math. , vol.47 , pp. 789-830
    • Chen, G.-Q.1    Levermore, C.D.2    Liu, T.-P.3
  • 10
    • 84968494859 scopus 로고
    • An error estimate for finite volume methods for conservation laws
    • B. Cockburn, F. Coquel and P. LeFloch, An error estimate for finite volume methods for conservation laws. Math. Comp. 64 (1994) 77-103.
    • (1994) Math. Comp. , vol.64 , pp. 77-103
    • Cockburn, B.1    Coquel, F.2    LeFloch, P.3
  • 11
    • 0003087464 scopus 로고
    • A posteriori error estimates for general numerical methods for scalar conservation laws
    • B. Cockburn and H. Gau, A posteriori error estimates for general numerical methods for scalar conservation laws. Math. Appl. Comp. 14 (1995) 37-47.
    • (1995) Math. Appl. Comp. , vol.14 , pp. 37-47
    • Cockburn, B.1    Gau, H.2
  • 12
    • 0002154319 scopus 로고    scopus 로고
    • Error estimates for finite element methods for scalar conservation laws
    • B. Cockburn and P.-A. Gremaud, Error estimates for finite element methods for scalar conservation laws. SIAM J. Numer. Anal. 33 (1996) 522-554.
    • (1996) SIAM J. Numer. Anal. , vol.33 , pp. 522-554
    • Cockburn, B.1    Gremaud, P.-A.2
  • 13
    • 84966261380 scopus 로고
    • The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case
    • B. Cockburn, S. Hou and C.-W. Shu, The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comp. 54 (1990) 545-581.
    • (1990) Math. Comp. , vol.54 , pp. 545-581
    • Cockburn, B.1    Hou, S.2    Shu, C.-W.3
  • 14
    • 0040456047 scopus 로고    scopus 로고
    • Advanced numerical approximation of nonlinear hyperbolic equations
    • A. Quarteroni (Ed.), Springer-Verlag
    • B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor, Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. A. Quarteroni (Ed.), Lect. Notes Math. 1697, Springer-Verlag (1998).
    • (1998) Lect. Notes Math. , vol.1697
    • Cockburn, B.1    Johnson, C.2    Shu, C.-W.3    Tadmor, E.4
  • 15
    • 45249130080 scopus 로고
    • TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems
    • B. Cockburn, S.Y. Lin and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84 (1989) 90-113.
    • (1989) J. Comput. Phys. , vol.84 , pp. 90-113
    • Cockburn, B.1    Lin, S.Y.2    Shu, C.-W.3
  • 16
    • 84966246413 scopus 로고
    • TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework
    • B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp. 52 (1989) 411-435.
    • (1989) Math. Comp. , vol.52 , pp. 411-435
    • Cockburn, B.1    Shu, C.-W.2
  • 17
    • 0001130290 scopus 로고    scopus 로고
    • Relaxation of energy and approximate riemann solvers for general pressure laws in fluid dynamics
    • F. Coquel and B. Pertharne, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 35 (1998) 2223-2249.
    • (1998) SIAM J. Numer. Anal. , vol.35 , pp. 2223-2249
    • Coquel, F.1    Pertharne, B.2
  • 19
    • 0007418740 scopus 로고    scopus 로고
    • A-posteriori error estimates for numerical approximations to scalar conservation laws: Schemes satisfying strong and weak entropy inequalities
    • L. Gosse and Ch. Makridakis, A-posteriori error estimates for numerical approximations to scalar conservation laws: schemes satisfying strong and weak entropy inequalities. IACM-FORTH Technical Report 98-4 (1998).
    • (1998) IACM-FORTH Technical Report , vol.98 , Issue.4
    • Gosse, L.1    Makridakis, Ch.2
  • 20
    • 0034437087 scopus 로고    scopus 로고
    • Two a posteriori error estimates for one dimensional scalar conservation laws
    • L. Gosse and Ch. Makridakis, Two a posteriori error estimates for one dimensional scalar conservation laws. SIAM J. Numer. Anal. 38 (2000) 964-988.
    • (2000) SIAM J. Numer. Anal. , vol.38 , pp. 964-988
    • Gosse, L.1    Makridakis, Ch.2
  • 21
    • 0039863145 scopus 로고    scopus 로고
    • Convergence of relaxation schemes to the equations of elastodynamics
    • to appear
    • L. Gosse and A. Tzavaras, Convergence of relaxation schemes to the equations of elastodynamics. Math. Comp. (to appear).
    • Math. Comp.
    • Gosse, L.1    Tzavaras, A.2
  • 22
    • 0000261039 scopus 로고
    • Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation
    • D. Jacobs, B. McKinney, M. Shearer, Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation. J. Differential Equations 116 (1995) 448-467.
    • (1995) J. Differential Equations , vol.116 , pp. 448-467
    • Jacobs, D.1    McKinney, B.2    Shearer, M.3
  • 23
    • 0001028401 scopus 로고
    • Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws
    • J. Jaffré, C. Johnson and A. Szepessy, Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci. 5 (1995) 367-386.
    • (1995) Math. Models Methods Appl. Sci. , vol.5 , pp. 367-386
    • Jaffré, J.1    Johnson, C.2    Szepessy, A.3
  • 24
    • 84990701264 scopus 로고
    • The relaxing schemes for systems of conservation laws in arbitrary space dimensions
    • 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.
    • (1995) Comm. Pure Appl. Math. , vol.48 , pp. 235-277
    • Jin, S.1    Xin, Z.2
  • 25
    • 84966200286 scopus 로고
    • On the convergence of a finite element method for a nonlinear hyperbolic conservation law
    • C. Johnson and A. Szepessy, On the convergence of a finite element method for a nonlinear hyperbolic conservation law. Math. Comp. 49 (1987) 427-444.
    • (1987) Math. Comp. , vol.49 , pp. 427-444
    • Johnson, C.1    Szepessy, A.2
  • 26
    • 84990662772 scopus 로고
    • Adaptive finite element methods for conservation laws. Part I: The general approach
    • C. Johnson and A. Szepessy, Adaptive finite element methods for conservation laws. Part I: The general approach. Comm. Pure Appl. Math. 48 (1995) 199-234.
    • (1995) Comm. Pure Appl. Math. , vol.48 , pp. 199-234
    • Johnson, C.1    Szepessy, A.2
  • 27
    • 0039863143 scopus 로고    scopus 로고
    • Finite volume relaxation schemes for multidimensional conservation laws
    • to appear
    • T. Katsaounis and Ch. Makridakis, Finite volume relaxation schemes for multidimensional conservation laws. Math. Comp. (to appear).
    • Math. Comp.
    • Katsaounis, T.1    Makridakis, Ch.2
  • 28
    • 0033478912 scopus 로고    scopus 로고
    • Convergence and error estimates of relaxation schemes for multidimensional conservation laws
    • M.A. Katsoulakis, G.T. Kossioris and Ch. Makridakis, Convergence and error estimates of relaxation schemes for multidimensional conservation laws. Comm. Partial Differential Equations 24 (1999) 395-424.
    • (1999) Comm. Partial Differential Equations , vol.24 , pp. 395-424
    • Katsoulakis, M.A.1    Kossioris, G.T.2    Makridakis, Ch.3
  • 29
    • 0000282942 scopus 로고    scopus 로고
    • Contractive relaxation systems and the scalar multidimensional conservation law
    • M.A. Katsoulakis and A.E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law. Comm. Partial Differential Equations 22 (1997) 195-233.
    • (1997) Comm. Partial Differential Equations , vol.22 , pp. 195-233
    • Katsoulakis, M.A.1    Tzavaras, A.E.2
  • 30
    • 0040016270 scopus 로고    scopus 로고
    • A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions
    • D. Kröner and M. Ohlberger, A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions. Math. Comp. 69 (2000) 25-39.
    • (2000) Math. Comp. , vol.69 , pp. 25-39
    • Kröner, D.1    Ohlberger, M.2
  • 31
    • 0347987814 scopus 로고    scopus 로고
    • New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations
    • A. Kurganov and E. Tadmor, New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282.
    • (2000) J. Comput. Phys. , vol.160 , pp. 241-282
    • Kurganov, A.1    Tadmor, E.2
  • 32
    • 0001161125 scopus 로고
    • Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation
    • N.N. Kuznetzov, Accuracy of some approximate methods for computing the weak solutions of a first-order quasi-linear equation. USSR Comput. Math. Math. Phys. 16 (1976) 105-119.
    • (1976) USSR Comput. Math. Math. Phys. , vol.16 , pp. 105-119
    • Kuznetzov, N.N.1
  • 33
    • 0000542076 scopus 로고
    • A study of numerical methods for hyperbolic conservation laws with stiff terms
    • R.J. LeVeque and H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff terms. J. Comput. Phys. 86 (1990) 187-210.
    • (1990) J. Comput. Phys. , vol.86 , pp. 187-210
    • LeVeque, R.J.1    Yee, H.C.2
  • 34
    • 21544449601 scopus 로고
    • Hyperbolic conservation laws with relaxation
    • T.-P. Liu, Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108 (1987) 153-175.
    • (1987) Comm. Math. Phys. , vol.108 , pp. 153-175
    • Liu, T.-P.1
  • 35
    • 0009343514 scopus 로고
    • A moving mesh numerical method for hyperbolic conservation laws
    • B.J. Lucier, A moving mesh numerical method for hyperbolic conservation laws. Math. Comp. 40 (1983) 91-106.
    • (1983) Math. Comp. , vol.40 , pp. 91-106
    • Lucier, B.J.1
  • 37
    • 0030490070 scopus 로고    scopus 로고
    • Convergence to equilibrium for the relaxation approximations of conservation laws
    • R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws. Comm. Pure Appl. Math. 8 (1996) 795 -823.
    • (1996) Comm. Pure Appl. Math. , vol.8 , pp. 795-823
    • Natalini, R.1
  • 38
    • 11544335821 scopus 로고    scopus 로고
    • A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws
    • R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Differential Equations 148 (1998) 292-317.
    • (1998) J. Differential Equations , vol.148 , pp. 292-317
    • Natalini, R.1
  • 39
    • 28144455569 scopus 로고
    • Non-oscillatory central differencing for hyperbolic conservation laws
    • H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463.
    • (1990) J. Comput. Phys. , vol.87 , pp. 408-463
    • Nessyahu, H.1    Tadmor, E.2
  • 40
    • 84966232121 scopus 로고
    • On the convergence of difference approximations to scalar conservation laws
    • S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51.
    • (1988) Math. Comp. , vol.50 , pp. 19-51
    • Osher, S.1    Tadmor, E.2
  • 41
    • 0003288339 scopus 로고    scopus 로고
    • An introduction to kinetic schemes for gas dynamics
    • An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde (Eds.), Springer-Verlag
    • B. Perthame, An introduction to kinetic schemes for gas dynamics, in: An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde (Eds.), Lect. Notes Comput. Sci. Eng. 5. Springer-Verlag (1998) 1-27.
    • (1998) Lect. Notes Comput. Sci. Eng. , vol.5 , pp. 1-27
    • Perthame, B.1
  • 43
    • 0039026206 scopus 로고    scopus 로고
    • 1 error bound for a semi-implicit difference scheme applied to a stiff system of conservation laws
    • 1 error bound for a semi-implicit difference scheme applied to a stiff system of conservation laws. SIAM J. Numer. Anal. 34 (1997) 1152-1166.
    • (1997) SIAM J. Numer. Anal. , vol.34 , pp. 1152-1166
    • Schroll, H.J.1    Tveito, A.2    Winther, R.3
  • 44
    • 0000305292 scopus 로고    scopus 로고
    • Relaxation semi linéaire et cinétique des systèmes de lois de conservation
    • D. Serre, Relaxation semi linéaire et cinétique des systèmes de lois de conservation. Ann. Inst. H. Poincaré, Anal. Non Linéaire 17 (2000) 169-192.
    • (2000) Ann. Inst. H. Poincaré, Anal. Non Linéaire , vol.17 , pp. 169-192
    • Serre, D.1
  • 45
    • 0000564951 scopus 로고
    • Total-variation-diminishing time discretizations
    • C.-W. Shu, Total-variation-diminishing time discretizations. SIAM J. Sci. Comput. 9 (1988) 1073-1084.
    • (1988) SIAM J. Sci. Comput. , vol.9 , pp. 1073-1084
    • Shu, C.-W.1
  • 46
    • 45449125925 scopus 로고
    • Efficient implementation of essentially nonoscillatory shock-capturing schemes
    • C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439-471.
    • (1988) J. Comput. Phys. , vol.77 , pp. 439-471
    • Shu, C.-W.1    Osher, S.2
  • 47
    • 0039835168 scopus 로고    scopus 로고
    • A dual graph-norm refinement indicator for finite volume approximations of the Euler equations
    • T. Sonar and E. Süli, A dual graph-norm refinement indicator for finite volume approximations of the Euler equations. Numer. Math. 78 (1998) 619-658.
    • (1998) Numer. Math. , vol.78 , pp. 619-658
    • Sonar, T.1    Süli, E.2
  • 48
    • 0002885640 scopus 로고    scopus 로고
    • A-postenori error analysis and adaptivity for finite element approximations of hyperbolic problems
    • An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde (Eds.), Springer-Verlag
    • E. Süli, A-postenori error analysis and adaptivity for finite element approximations of hyperbolic problems, in: An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde (Eds.), Lect. Notes Comput. Sci. Eng. 5, Springer-Verlag (1998) 123-194.
    • (1998) Lect. Notes Comput. Sci. Eng. , vol.5 , pp. 123-194
    • Süli, E.1
  • 49
    • 84968514036 scopus 로고
    • Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions
    • A. Szepessy, Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions. Math. Comp. 53 (1989) 527-545.
    • (1989) Math. Comp. , vol.53 , pp. 527-545
    • Szepessy, A.1
  • 50
    • 0001784203 scopus 로고    scopus 로고
    • Viscosity and relaxation approximation for hyperbolic systems of conservation laws
    • An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde (Eds.), Springer-Verlag
    • A. Tzavaras, Viscosity and relaxation approximation for hyperbolic systems of conservation laws, in: An introduction to recent developments in theory and numerics for conservation laws, D. Kröner, M. Ohlberger and C. Rohde (Eds.), Lect. Notes Comput. Sci. Eng. 5, Springer-Verlag (1998) 73-122.
    • (1998) Lect. Notes Comput. Sci. Eng. , vol.5 , pp. 73-122
    • Tzavaras, A.1
  • 51
    • 0033425513 scopus 로고    scopus 로고
    • Materials with internal variables and relaxation to conservation laws
    • A. Tzavaras, Materials with internal variables and relaxation to conservation laws. Arch. Rational Mech. Anal. 146 (1999) 199-155.
    • (1999) Arch. Rational Mech. Anal. , vol.146 , pp. 199-1155
    • Tzavaras, A.1


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