We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whose leaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate the computational efficiency together with the convergence properties.
@article{M2AN_2008__42_4_535_0,
author = {B\"urger, Raimund and Ruiz, Ricardo and Schneider, Kai and Sep\'ulveda, Mauricio},
title = {Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {42},
year = {2008},
pages = {535-563},
doi = {10.1051/m2an:2008016},
mrnumber = {2437773},
zbl = {1147.65066},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2008__42_4_535_0}
}
Bürger, Raimund; Ruiz, Ricardo; Schneider, Kai; Sepúlveda, Mauricio. Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 535-563. doi : 10.1051/m2an:2008016. http://gdmltest.u-ga.fr/item/M2AN_2008__42_4_535_0/
[1] and , An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1-102. | MR 2009692 | Zbl 1105.65349
[2] , , and , Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput. 15 (1994) 127-138. | MR 1257158 | Zbl 0793.65072
[3] and , Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35 (1998) 2298-2316. | MR 1655847 | Zbl 0921.65070
[4] and , Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53 (1984) 484-512. | MR 739112 | Zbl 0536.65071
[5] , , and , Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64 (2003) 41-80. | MR 2029124 | Zbl 1047.35071
[6] and , On some upwind schemes for the phenomenological sedimentation-consolidation model. J. Eng. Math. 41 (2001) 145-166. | MR 1866604 | Zbl 1128.76341
[7] and , On a diffusively corrected kinematic-wave traffic model with changing road surface conditions. Math. Models Meth. Appl. Sci. 13 (2003) 1767-1799. | MR 2032211 | Zbl 1055.35071
[8] , and , On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl. 247 (2000) 517-556. | MR 1769093 | Zbl 0961.35078
[9] , , and , Well-posedness in and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. Numer. Math. 97 (2004) 25-65. | MR 2045458 | Zbl 1053.76047
[10] , and , A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units. SIAM J. Appl. Math. 65 (2005) 882-940. | MR 2136036 | Zbl 1089.76061
[11] , and , A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modelling sedimentation-consolidation processes. Math. Comp. 75 (2006) 91-112. | MR 2176391 | Zbl 1082.65081
[12] , and , On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels. Appl. Numer. Math. 56 (2006) 1397-1417. | MR 2245464 | Zbl 1103.65091
[13] , and , Multiresolution schemes for strongly degenerate parabolic equations in one space dimension. Numer. Meth. Partial Diff. Equ. 23 (2007) 706-730. | MR 2310269 | Zbl 1114.65120
[14] , , and , Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux. J. Eng. Math. 60 (2008) 365-385. | MR 2396490 | Zbl 1137.65393
[15] , Entropy solutions for nonlinear degenerate problems. Arch. Rat. Mech. Anal. 147 (1999) 269-361. | MR 1709116 | Zbl 0935.35056
[16] , and , Multiresolution-based adaptive schemes for hyperbolic conservation laws, in Adaptive Mesh Refinement-Theory and Applications, T. Plewa, T. Linde and V.G. Weiss Eds., Lect. Notes Computat. Sci. Engrg. 41, Springer-Verlag, Berlin (2003) 137-159. | Zbl 1065.65118
[17] , , and , Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comp. 72 (2002) 183-225. | MR 1933818 | Zbl 1010.65035
[18] and , Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1-21. | MR 551288 | Zbl 0423.65052
[19] and , Scientific Computing with Ordinary Differential Equations. Springer-Verlag, New York (2002). | MR 1912409 | Zbl 1001.65071
[20] , Speed/flow relationships within an urban area. Traffic Eng. Control 8 (1966) 393-396.
[21] , and , An adaptive multiresolution method for parabolic PDEs with time step control. ESAIM: Proc. 16 (2007) 181-194. | MR 2312857 | Zbl 1206.65228 | Zbl pre05213236
[22] , , and , An adaptive multiresolution scheme with local time-stepping for evolutionary PDEs. J. Comput. Phys. 227 (2008) 3758-3780. | MR 2403866 | Zbl 1139.65060
[23] and , One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321-351. | MR 606500 | Zbl 0469.65067
[24] and , Numerical solution of reservoir flow models based on large time step operator splitting methods, in Filtration in Porous Media and Industrial Application, M.S. Espedal, A. Fasano and A. Mikelić Eds., Springer-Verlag, Berlin (2000) 9-77. | MR 1816143 | Zbl 1077.76546
[25] and , Monotone difference approximations of solutions to degenerate convection-diffusion equations. SIAM J. Numer. Anal. 37 (2000) 1838-1860. | MR 1766850 | Zbl 0985.65100
[26] , , and , Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41-82. | MR 1917365 | Zbl 1005.65099
[27] , Low order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. Computing 6 (1970) 61-71. | Zbl 0217.53001
[28] and , Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). | MR 1410987 | Zbl 0860.65075
[29] , An analysis of traffic flow. Oper. Res. 7 (1959) 79-85. | MR 101166
[30] , and , Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edn., Springer-Verlag, Berlin (1993). | MR 1227985 | Zbl 0789.65048
[31] , Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48 (1995) 1305-1342. | MR 1369391 | Zbl 0860.65078
[32] , and , On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. 29 (1976) 297-322. | MR 413526 | Zbl 0351.76070
[33] and , Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: M2AN 35 (2001) 239-269. | Numdam | MR 1825698 | Zbl 1032.76048
[34] , and , Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal. 22 (2002) 623-664. | MR 1937244 | Zbl 1014.65073
[35] , and , stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vid. Selsk. (2003) 1-49. | MR 2024741 | Zbl 1036.35104
[36] , First order quasilinear equations in several independent space variables. Math. USSR Sb. 10 (1970) 217-243. | Zbl 0215.16203
[37] , Accuracy of some approximate methods for computing the weak solutions of a first order quasilinear equation. USSR Comp. Math. Math. Phys. 16 (1976) 105-119. | Zbl 0381.35015
[38] and , On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London Ser. A 229 (1955) 317-345. | MR 72606 | Zbl 0064.20906
[39] and , Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods. SIAM J. Numer. Anal. 41 (2003) 2262-2293. | MR 2034615 | Zbl 1058.35127
[40] , Adaptive Multiscale Schemes for Conservation Laws. Springer-Verlag, Berlin (2003). | MR 1952371 | Zbl 1016.76004
[41] and , Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J. Sci. Comp. 30 (2007) 493-531. | MR 2295481 | Zbl 1110.76037
[42] , Traveling-wave solutions of the diffusively corrected kinematic-wave model. Math. Comp. Modelling 35 (2002) 561-579. | MR 1884018 | Zbl 0994.90031
[43] , Shock waves on the highway. Oper. Res. 4 (1956) 42-51. | MR 75522
[44] and , An adaptive multiresolution method for combustion problems: Application to flame ball-vortex interaction. Comput. Fluids 34 (2005) 817-831. | Zbl 1134.80304
[45] , , and , A conservative fully adaptive multiresolution algorithm for parabolic conservation laws. J. Comput. Phys. 188 (2003) 493-523. | MR 1985307 | Zbl 1022.65093
[46] , Métodos de Multiresolución y su Aplicación a un Problema de Ingeniería. Tesis para optar al título de Ingeniero Matemático, Universidad de Concepción, Chile (2005).
[47] , Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor, in Lecture Notes in Mathematics 1697, A. Quarteroni Ed., Springer-Verlag, Berlin (1998) 325-432. | MR 1728856 | Zbl 0927.65111
[48] and , Numerische Mathematik 2. 3rd Edn., Springer-Verlag, Berlin (1990). | MR 1100482 | Zbl 0693.65001
[49] and , An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003). | MR 2006500 | Zbl 1033.65001
[50] , Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000) 681-698. | MR 1770068 | Zbl 0972.65060
[51] , A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197-1218. | MR 1870839 | Zbl 1055.65104