# 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.

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

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topArvanitis, 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] D. Aregba-Driollet and R. Natalini, Convergence of relaxation schemes for conservation laws. Appl. Anal. 61 (1996) 163–193. Zbl0872.65086
- [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] I. Babuška, The adaptive finite element method. TICAM Forum Notes no 7, University of Texas at Austin (1997).
- [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] 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] 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] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York (1994). Zbl0804.65101MR1278258
- [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] L. Gosse and A. Tzavaras, Convergence of relaxation schemes to the equations of elastodynamics. Math. Comp. (to appear). Zbl0964.35097MR1813140
- [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] 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] 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] 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] 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] T. Katsaounis and Ch. Makridakis, Finite volume relaxation schemes for multidimensional conservation laws. Math. Comp. (to appear). Zbl0967.65094MR1681104
- [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] 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] 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] 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] 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] 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] T.-P. Liu, Hyperbolic conservation laws with relaxation. Comm. Math. Phys. 108 (1987) 153–175. Zbl0633.35049
- [35] B.J. Lucier, A moving mesh numerical method for hyperbolic conservation laws. Math. Comp. 40 (1983) 91–106. Zbl0592.65062
- [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] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws. Comm. Pure Appl. Math. 8 (1996) 795–823. Zbl0872.35064
- [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] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87 (1990) 408–463. Zbl0697.65068
- [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] 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] 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] 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] 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] C.-W. Shu, Total-variation-diminishing time discretizations. SIAM J. Sci. Comput. 9 (1988) 1073–1084. Zbl0662.65081
- [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] 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] 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] 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] 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] A. Tzavaras, Materials with internal variables and relaxation to conservation laws. Arch. Rational Mech. Anal. 146 (1999) 129–155. Zbl0973.74005

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