Adaptive finite element relaxation schemes for hyperbolic conservation laws

Christos Arvanitis; Theodoros Katsaounis[1]; Charalambos Makridakis

  • [1] Department of Applied Mathematics, University of Crete, Heraklion 71409, Greece.

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 1, page 17-33
  • ISSN: 0764-583X

Abstract

top
We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar problems and the system of elastodynamics.

How to cite

top

Arvanitis, Christos, Katsaounis, Theodoros, and Makridakis, Charalambos. "Adaptive finite element relaxation schemes for hyperbolic conservation laws." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.1 (2001): 17-33. <http://eudml.org/doc/194042>.

@article{Arvanitis2001,
abstract = {We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar problems and the system of elastodynamics.},
affiliation = {Department of Applied Mathematics, University of Crete, Heraklion 71409, Greece.},
author = {Arvanitis, Christos, Katsaounis, Theodoros, Makridakis, Charalambos},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {conservation laws; finite elements; adaptive methods; semidiscretization; consistency; adaptive refinement; shock; mesh coarsening; Runge-Kutta methods; Burgers equation; nonlinear elastodynamics},
language = {eng},
number = {1},
pages = {17-33},
publisher = {EDP-Sciences},
title = {Adaptive finite element relaxation schemes for hyperbolic conservation laws},
url = {http://eudml.org/doc/194042},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Arvanitis, Christos
AU - Katsaounis, Theodoros
AU - Makridakis, Charalambos
TI - Adaptive finite element relaxation schemes for hyperbolic conservation laws
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 1
SP - 17
EP - 33
AB - We propose and study semidiscrete and fully discrete finite element schemes based on appropriate relaxation models for systems of Hyperbolic Conservation Laws. These schemes are using piecewise polynomials of arbitrary degree and their consistency error is of high order. The methods are combined with an adaptive strategy that yields fine mesh in shock regions and coarser mesh in the smooth parts of the solution. The computational performance of these methods is demonstrated by considering scalar problems and the system of elastodynamics.
LA - eng
KW - conservation laws; finite elements; adaptive methods; semidiscretization; consistency; adaptive refinement; shock; mesh coarsening; Runge-Kutta methods; Burgers equation; nonlinear elastodynamics
UR - http://eudml.org/doc/194042
ER -

References

top
  1. [1] D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws. Appl. Anal. 61 (1996) 163–193. Zbl0872.65086
  2. [2] D. Aregba-Driollet and R. Natalini, Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973–2004. Zbl0964.65096
  3. [3] I. Babuška, The adaptive finite element method. TICAM Forum Notes no 7, University of Texas at Austin (1997). 
  4. [4] 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. Zbl0572.65095
  5. [5] M. Berger and R. LeVeque, Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35 (1998) 2298–2316. Zbl0921.65070
  6. [6] 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. Zbl0957.82028
  7. [7] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). Zbl0804.65101MR1278258
  8. [8] 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. Zbl0438.76059
  9. [9] 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. Zbl0806.35112
  10. [10] B. Cockburn, F. Coquel and P. LeFloch, An error estimate for finite volume methods for conservation laws. Math. Comp. 64 (1994) 77–103. Zbl0855.65103
  11. [11] B. Cockburn and H. Gau, A posteriori error estimates for general numerical methods for scalar conservation laws. Math. Appl. Comp. 14 (1995) 37–47. Zbl0834.65091
  12. [12] B. Cockburn and P.-A. Gremaud, Error estimates for finite element methods for scalar conservation laws. SIAM J. Numer. Anal. 33 (1996) 522–554. Zbl0861.65077
  13. [13] 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. Zbl0695.65066
  14. [14] 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). Zbl0904.00047MR1729305
  15. [15] 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. Zbl0677.65093
  16. [16] 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. Zbl0662.65083
  17. [17] F. Coquel and B. Perthame, Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 35 (1998) 2223–2249. Zbl0960.76051
  18. [18] K. Dekker and J.D. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, North-Holland, Amsterdam (1984). Zbl0571.65057MR774402
  19. [19] 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). 
  20. [20] L. Gosse and Ch. Makridakis, Two a posteriori error estimates for one dimensional scalar conservation laws. SIAM J. Numer. Anal. 38 (2000) 964–988. Zbl0980.65096
  21. [21] L. Gosse and A. Tzavaras, Convergence of relaxation schemes to the equations of elastodynamics. Math. Comp. (to appear). Zbl0964.35097MR1813140
  22. [22] D. Jacobs, B. McKinney, M. Shearer, Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation. J. Differential Equations 116 (1995) 448–467. Zbl0820.35118
  23. [23] 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. Zbl0834.65089
  24. [24] 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. Zbl0826.65078MR1322811
  25. [25] 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. Zbl0634.65075
  26. [26] 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. Zbl0826.65088
  27. [27] T. Katsaounis and Ch. Makridakis, Finite volume relaxation schemes for multidimensional conservation laws. Math. Comp. (to appear). Zbl0967.65094MR1681104
  28. [28] 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. Zbl0926.35088
  29. [29] M.A. Katsoulakis and A.E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law. Comm. Partial Differential Equations 22 (1997) 195–233. Zbl0884.35093
  30. [30] 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. Zbl0934.65102
  31. [31] 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. Zbl0987.65085
  32. [32] 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. Zbl0381.35015
  33. [33] 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. Zbl0682.76053
  34. [34] T.-P. Liu, Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108 (1987) 153–175. Zbl0633.35049
  35. [35] B.J. Lucier, A moving mesh numerical method for hyperbolic conservation laws. Math. Comp. 40 (1983) 91–106. Zbl0592.65062
  36. [36] J.J.H. Miller, E. O’Riordan and G.I. Shishkin, Fitted numerical methods for singular perturbation problems. Error estimates in the maximum norm for linear problems in one and two dimensions. World Scientific Publishing Co., River Edge, NJ (1996). Zbl0915.65097
  37. [37] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws. Comm. Pure Appl. Math. 8 (1996) 795–823. Zbl0872.35064
  38. [38] R. Natalini, A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Differential Equations 148 (1998) 292–317. Zbl0911.35073
  39. [39] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. Zbl0697.65068
  40. [40] S. Osher and E. Tadmor, On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19–51. Zbl0637.65091
  41. [41] 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. Zbl0969.76055
  42. [42] H.-G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems. Springer-Verlag, Berlin (1996). Zbl0844.65075MR1477665
  43. [43] H.J. Schroll, A. Tveito and R. Winther, An L 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. Zbl0873.65090
  44. [44] D. Serre, Relaxation semi linéaire et cinétique des systèmes de lois de conservation. Ann. Inst. H. Poincaré, Anal. Non Linéaire 17 (2000) 169–192. Zbl0963.35117
  45. [45] C.-W. Shu, Total-variation-diminishing time discretizations. SIAM J. Sci. Comput. 9 (1988) 1073–1084. Zbl0662.65081
  46. [46] C.-W. Shu and S. Osher, Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439–471. Zbl0653.65072
  47. [47] 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. Zbl0902.76083
  48. [48] E. Süli, A-posteriori 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 . Zbl0927.65117
  49. [49] 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. Zbl0679.65072
  50. [50] 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. Zbl0969.74006
  51. [51] A. Tzavaras, Materials with internal variables and relaxation to conservation laws. Arch. Rational Mech. Anal. 146 (1999) 129–155. Zbl0973.74005

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.