We propose a 1D adaptive numerical scheme for hyperbolic conservation laws based on the numerical density of entropy production (the amount of violation of the theoretical entropy inequality). This density is used as an a posteriori error which provides information if the mesh should be refined in the regions where discontinuities occur or coarsened in the regions where the solution remains smooth. As due to the Courant-Friedrich-Levy stability condition the time step is restricted and leads to time consuming simulations, we propose a local time stepping algorithm. We also use high order time extensions applying the Adams-Bashforth time integration technique as well as the second order linear reconstruction in space. We numerically investigate the efficiency of the scheme through several test cases: Sod’s shock tube problem, Lax’s shock tube problem and the Shu-Osher test problem.
@article{bwmeta1.element.doi-10_2478_s11533-013-0252-6,
author = {Mehmet Ersoy and Fr\'ed\'eric Golay and Lyudmyla Yushchenko},
title = {Adaptive multiscale scheme based on numerical density of entropy production for conservation laws},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {1392-1415},
zbl = {06201581},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0252-6}
}
Mehmet Ersoy; Frédéric Golay; Lyudmyla Yushchenko. Adaptive multiscale scheme based on numerical density of entropy production for conservation laws. Open Mathematics, Tome 11 (2013) pp. 1392-1415. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-013-0252-6/
[1] Allahviranloo T., Ahmady N., Ahmady E., Numerical solution of fuzzy differential equations by predictor-corrector method, Inform. Sci., 2007, 177(7), 1633–1647 http://dx.doi.org/10.1016/j.ins.2006.09.015 | Zbl 1183.65090
[2] Altmann C., Belat T., Gutnic M., Helluy P., Mathis H., Sonnendrücker É., Angulo W., Hérard J.-M., A local timestepping discontinuous Galerkin algorithm for the MHD system, In: CEMRACS 2008 - Modelling and Numerical Simulation of Complex Fluids, Marseille, July 21–August 29, 2008, ESAIM Proc., 28, EDP Sciences, Les Ulis, 2009, 33–54 | Zbl 1176.76058
[3] Berger M.J., Oliger J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 1984, 53(3), 484–512 http://dx.doi.org/10.1016/0021-9991(84)90073-1 | Zbl 0536.65071
[4] Cockburn B., Gremaud P.-A., A priori error estimates for numerical methods for scalar conservation laws. Part II: Flux-splitting monotone schemes on irregular Cartesian grids, Math. Comp., 1997, 66(218), 547–572 http://dx.doi.org/10.1090/S0025-5718-97-00838-7 | Zbl 0866.65061
[5] Croisille J.-P., Contribution à l’Étude Théorique et à l’Approximation par Éléments Finis du Système Hyperbolique de la Dynamique des Gaz Multidimensionnelle et Multiespèces, PhD thesis, Université de Paris VI, 1991
[6] Eymard R., Gallouët T., Herbin R., Finite Volume Methods, In: Handbook of Numerical Analysis, VII, Handb. Numer. Anal., VII, North-Holland, Amsterdam, 2000, 713–1020 | Zbl 0981.65095
[7] Gallouët T., Hérard J.-M., Seguin N., Some recent finite volume schemes to compute Euler equations using real gas EOS, Internat. J. Numer. Methods Fluids, 2002, 39(12), 1073–1138 http://dx.doi.org/10.1002/fld.346 | Zbl 1053.76044
[8] Godlewski E., Raviart P.-A., Numerical Approximation of Hyperbolic Systems of Conservation Laws, Appl. Math. Sci., 118, Springer, New York, 1996 | Zbl 0860.65075
[9] Golay F., Numerical entropy production and error indicator for compressible flows, Comptes Rendus Mécanique, 2009, 337(4), 233–237 http://dx.doi.org/10.1016/j.crme.2009.04.004 | Zbl 1173.76402
[10] Guermond J.-L., Pasquetti R., Popov B., Entropy viscosity method for nonlinear conservation laws, J. Comput. Phys., 2011, 230(11), 4248–4267 http://dx.doi.org/10.1016/j.jcp.2010.11.043 | Zbl 1220.65134
[11] Hairer E., Nørsett S.P., Wanner G., Solving Ordinary Differential Equations. I, 2nd ed., Springer Ser. Comput. Math., 8, Springer, Berlin, 1993 | Zbl 0789.65048
[12] Houston P., Mackenzie J.A., Süli E., Warnecke G., A posteriori error analysis for numerical approximations of Friedrichs systems, Numer. Math., 1999, 82(3), 433–470 http://dx.doi.org/10.1007/s002110050426 | Zbl 0935.65096
[13] Jenny P., Lee S.H., Tchelepi H.A., Adaptive multiscale finite-volume method for multiphase flow and transport in porous media, Multiscale Model. Simul., 2005, 3(1), 50–64 http://dx.doi.org/10.1137/030600795 | Zbl 1160.76372
[14] Karni S., Kurganov A., Local error analysis for approximate solutions of hyperbolic conservation laws, Adv. Comput. Math., 2005, 22(1), 79–99 http://dx.doi.org/10.1007/s10444-005-7099-8 | Zbl 1127.65070
[15] Karni S., Kurganov A., Petrova G., A smoothness indicator for adaptive algorithms for hyperbolic systems, J. Comput. Phys., 2002, 178(2), 323–341 http://dx.doi.org/10.1006/jcph.2002.7024 | Zbl 0998.65092
[16] Müller S., Stiriba Y., Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping, J. Sci. Comput., 2007, 30(3), 493–531 http://dx.doi.org/10.1007/s10915-006-9102-z | Zbl 1110.76037
[17] Osher S., Sanders R., Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comp., 1983, 41(164), 321–336 http://dx.doi.org/10.1090/S0025-5718-1983-0717689-8 | Zbl 0592.65068
[18] Puppo G., Numerical entropy production on shocks and smooth transitions, J. Sci. Comput., 2002, 17(1–4), 263–271 http://dx.doi.org/10.1023/A:1015117118157 | Zbl 1001.76070
[19] Puppo G., Numerical entropy production for central schemes, SIAM J. Sci. Comput., 2004, 25(4), 1382–1415 http://dx.doi.org/10.1137/S1064827502386712 | Zbl 1061.65094
[20] Puppo G., Semplice M., Numerical entropy and adaptivity for finite volume schemes, Commun. Comput. Phys., 2011, 10(5), 1132–1160
[21] Shu C.-W., Osher S., Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 1988, 77(2), 439–471 http://dx.doi.org/10.1016/0021-9991(88)90177-5 | Zbl 0653.65072
[22] Simeoni C., Remarks on the consistency of upwind source at interface schemes on nonuniform grids, J. Sci. Comput., 2011, 48(1–3), 333–338 http://dx.doi.org/10.1007/s10915-010-9442-6 | Zbl 1221.65239
[23] Sod G.A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 1978, 27(1), 1–31 http://dx.doi.org/10.1016/0021-9991(78)90023-2
[24] Sonar T., Hannemann V., Hempel D. Dynamic adaptivity and residual control in unsteady compressible flow computation, In: Theory and Numerical Methods for Initial-Boundary Value Problems, Math. Comput. Modelling, 1994, 20(10–11), 201–213 http://dx.doi.org/10.1016/0895-7177(94)90178-3 | Zbl 0816.76073
[25] Tan Z., Zhang Z., Huang Y., Tang T., Moving mesh methods with locally varying time steps, J. Comput. Phys., 2004, 200(1), 347–367 http://dx.doi.org/10.1016/j.jcp.2004.04.007 | Zbl 1054.65099
[26] Tang H., Warnecke G., A class of high resolution difference schemes for nonlinear Hamilton-Jacobi equations with varying time and space grids, SIAM J. Sci. Comput., 2005, 26(4), 1415–1431 http://dx.doi.org/10.1137/S1064827503428126 | Zbl 1079.65092
[27] Toro E.F., Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd ed., Springer, Berlin, 1999 http://dx.doi.org/10.1007/978-3-662-03915-1 | Zbl 0923.76004
[28] Zhang X.D., Trépanier J.-Y., Camarero R., A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws, Comput. Methods Appl. Engrg., 2000, 185(1), 1–19 http://dx.doi.org/10.1016/S0045-7825(99)00099-7 | Zbl 0954.65071