Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T02:40:53.780Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptive Finite Element Relaxation Schemes for Hyperbolic Conservation Laws

Published online by Cambridge University Press:  15 April 2002

Christos Arvanitis ,
Theodoros Katsaounis  and
Charalambos Makridakis
Christos Arvanitis
Affiliation:
Department of Mathematics, University of Crete, Heraklion 71409, Greece. Institute for Applied and Computational Mathematics, FORTH, Heraklion 71110, Greece.
Theodoros Katsaounis
Affiliation:
Institute for Applied and Computational Mathematics, FORTH, Heraklion 71110, Greece. Department of Applied Mathematics, University of Crete, Heraklion 71409, Greece.
Charalambos Makridakis
Affiliation:
Institute for Applied and Computational Mathematics, FORTH, Heraklion 71110, Greece.
Get access

Abstract

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

Keywords

Conservation lawsfinite elementsadaptive methods.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 1 , January 2001 , pp. 17 - 33
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aregba-Driollet, D. and Natalini, R., Convergence of relaxation schemes for conservation laws. Appl. Anal. 61 (1996) 163-193. CrossRef
Aregba-Driollet, D. and Natalini, R., Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37 (2000) 1973-2004. CrossRef
I. Babuska, The adaptive finite element method. TICAM Forum Notes no 7, University of Texas at Austin (1997).
Babuska, I. and Gui, W., Basic principles of feedback and adaptive approaches in the finite element method. Comput. Methods Appl. Mech. Engrg. 55 (1986) 27-42.
Berger, M. and LeVeque, R., Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35 (1998) 2298-2316. CrossRef
Bouchut, F., Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Statist. Phys. 95 (1999) 113-170. CrossRef
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994).
Caflisch, R.E. and Papanicolaou, G.C., The fluid dynamical limit of a nonlinear model Boltzmann equation. Comm. Pure Appl. Math. 32 (1979) 589-616. CrossRef
Chen, G.-Q., Levermore, C.D. and Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy. Comm. Pure Appl. Math. 47 (1994) 789-830.
Cockburn, B., Coquel, F. and LeFloch, P., An error estimate for finite volume methods for conservation laws. Math. Comp. 64 (1994) 77-103. CrossRef
Cockburn, B. and Gau, H., A posteriori error estimates for general numerical methods for scalar conservation laws. Math. Appl. Comp. 14 (1995) 37-47.
Cockburn, B. and Gremaud, P.-A., Error estimates for finite element methods for scalar conservation laws. SIAM J. Numer. Anal. 33 (1996) 522-554. CrossRef
Cockburn, B., Hou, S. and Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comp. 54 (1990) 545-581.
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).
Cockburn, B., Lin, S.Y. and Shu, C.-W., Runge-Kutta, TVB local projection discontinuous Galerkin finite element method for conservation laws. III. One-dimensional systems. J. Comput. Phys. 84 (1989) 90-113. CrossRef
Cockburn, B. and Shu, C.-W., Runge-Kutta, TVB local projection discontinuous Galerkin finite element method for conservation laws. II. General framework. Math. Comp. 52 (1989) 411-435.
Coquel, F. and Perthame, B., Relaxation of energy and approximate Riemann solvers for general pressure laws in fluid dynamics. SIAM J. Numer. Anal. 35 (1998) 2223-2249. CrossRef
K. Dekker and J.D. Verwer, Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monographs, North-Holland, Amsterdam (1984).
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).
Gosse, L. and Makridakis, Ch., Two a posteriori error estimates for one dimensional scalar conservation laws. SIAM J. Numer. Anal. 38 (2000) 964-988. CrossRef
L. Gosse and A. Tzavaras, Convergence of relaxation schemes to the equations of elastodynamics. Math. Comp. (to appear).
Jacobs, D., McKinney, B., Shearer, M., Travelling wave solutions of the modified Korteweg-de Vries-Burgers equation. J. Differential Equations 116 (1995) 448-467. CrossRef
Jaffré, J., Johnson, C. and Szepessy, A., Convergence of the discontinuous Galerkin finite element method for hyperbolic conservation laws. Math. Models Methods Appl. Sci. 5 (1995) 367-386.
Jin, S. and Xin, Z., The relaxing schemes for systems of conservation laws in arbitrary space dimensions. Comm. Pure Appl. Math. 48 (1995) 235-277. CrossRef
Johnson, C. and Szepessy, A., On the convergence of a finite element method for a nonlinear hyperbolic conservation law. Math. Comp. 49 (1987) 427-444. CrossRef
Johnson, C. and Szepessy, A., Adaptive finite element methods for conservation laws. Part I: The general approach. Comm. Pure Appl. Math. 48 (1995) 199-234. CrossRef
T. Katsaounis and Ch. Makridakis, Finite volume relaxation schemes for multidimensional conservation laws. Math. Comp. (to appear).
Katsoulakis, M.A., Kossioris, G.T. and Makridakis, Ch., Convergence and error estimates of relaxation schemes for multidimensional conservation laws. Comm. Partial Differential Equations 24 (1999) 395-424. CrossRef
Katsoulakis, M.A. and Tzavaras, A.E., Contractive relaxation systems and the scalar multidimensional conservation law. Comm. Partial Differential Equations 22 (1997) 195-233.
Kröner, D. and Ohlberger, M., A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multidimensions. Math. Comp. 69 (2000) 25-39. CrossRef
Kurganov, A. and Tadmor, E., New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J. Comput. Phys. 160 (2000) 241-282. CrossRef
Kuznetzov, N.N., 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.
LeVeque, R.J. and Yee, H.C., A study of numerical methods for hyperbolic conservation laws with stiff terms. J. Comput. Phys. 86 (1990) 187-210. CrossRef
Liu, T.-P., Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108 (1987) 153-175. CrossRef
Lucier, B.J., A moving mesh numerical method for hyperbolic conservation laws. Math. Comp. 40 (1983) 91-106.
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).
R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws. Comm. Pure Appl. Math. 8 (1996) 795-823.
Natalini, R., A discrete kinetic approximation of entropy solutions to multidimensional scalar conservation laws. J. Differential Equations 148 (1998) 292-317. CrossRef
Nessyahu, H. and Tadmor, E., Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408-463. CrossRef
Osher, S. and Tadmor, E., On the convergence of difference approximations to scalar conservation laws. Math. Comp. 50 (1988) 19-51. CrossRef
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.
H.-G. Roos, M. Stynes and L. Tobiska, Numerical methods for singularly perturbed differential equations. Convection-diffusion and flow problems. Springer-Verlag, Berlin (1996).
Schroll, H.J., Tveito, A. and Winther, R., 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. CrossRef
D. Serre, Relaxation semi linéaire et cinétique des systèmes de lois de conservation. Ann. Inst. H. Poincaré, Anal. Non Linéaire 17 (2000) 169-192.
Shu, C.-W., Total-variation-diminishing time discretizations. SIAM J. Sci. Comput. 9 (1988) 1073-1084. CrossRef
Shu, C.-W. and Osher, S., Efficient implementation of essentially nonoscillatory shock-capturing schemes. J. Comput. Phys. 77 (1988) 439-471. CrossRef
Sonar, T. and Süli, E., A dual graph-norm refinement indicator for finite volume approximations of the Euler equations. Numer. Math. 78 (1998) 619-658. CrossRef
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 .
Szepessy, A., 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. CrossRef
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.
Tzavaras, A., Materials with internal variables and relaxation to conservation laws. Arch. Rational Mech. Anal. 146 (1999) 129-155. CrossRef