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.
Mots-clés : degenerate parabolic equation, adaptive multiresolution scheme, monotone scheme, upwind difference scheme, boundary conditions, entropy solution
@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: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {535--563},
publisher = {EDP-Sciences},
volume = {42},
number = {4},
year = {2008},
doi = {10.1051/m2an:2008016},
mrnumber = {2437773},
zbl = {1147.65066},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2008016/}
}
TY - JOUR AU - Bürger, Raimund AU - Ruiz, Ricardo AU - Schneider, Kai AU - Sepúlveda, Mauricio TI - Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 535 EP - 563 VL - 42 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008016/ DO - 10.1051/m2an:2008016 LA - en ID - M2AN_2008__42_4_535_0 ER -
%0 Journal Article %A Bürger, Raimund %A Ruiz, Ricardo %A Schneider, Kai %A Sepúlveda, Mauricio %T Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 535-563 %V 42 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008016/ %R 10.1051/m2an:2008016 %G en %F 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: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 535-563. doi : 10.1051/m2an:2008016. http://www.numdam.org/articles/10.1051/m2an:2008016/
[1] and , An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1-102. | MR | Zbl
[2] , , and , Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput. 15 (1994) 127-138. | MR | Zbl
[3] and , Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35 (1998) 2298-2316. | MR | Zbl
[4] and , Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53 (1984) 484-512. | MR | Zbl
[5] , , and , Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64 (2003) 41-80. | MR | Zbl
[6] and , On some upwind schemes for the phenomenological sedimentation-consolidation model. J. Eng. Math. 41 (2001) 145-166. | MR | Zbl
[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 | Zbl
[8] , and , On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl. 247 (2000) 517-556. | MR | Zbl
[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 | Zbl
[10] , and , A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units. SIAM J. Appl. Math. 65 (2005) 882-940. | MR | Zbl
[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 | Zbl
[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 | Zbl
[13] , and , Multiresolution schemes for strongly degenerate parabolic equations in one space dimension. Numer. Meth. Partial Diff. Equ. 23 (2007) 706-730. | MR | Zbl
[14] , , and , Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux. J. Eng. Math. 60 (2008) 365-385. | MR | Zbl
[15] , Entropy solutions for nonlinear degenerate problems. Arch. Rat. Mech. Anal. 147 (1999) 269-361. | MR | Zbl
[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
[17] , , and , Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comp. 72 (2002) 183-225. | MR | Zbl
[18] and , Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1-21. | MR | Zbl
[19] and , Scientific Computing with Ordinary Differential Equations. Springer-Verlag, New York (2002). | MR | Zbl
[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 | Zbl
[22] , , and , An adaptive multiresolution scheme with local time-stepping for evolutionary PDEs. J. Comput. Phys. 227 (2008) 3758-3780. | MR | Zbl
[23] and , One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321-351. | MR | Zbl
[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 | Zbl
[25] and , Monotone difference approximations of solutions to degenerate convection-diffusion equations. SIAM J. Numer. Anal. 37 (2000) 1838-1860. | MR | Zbl
[26] , , and , Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41-82. | MR | Zbl
[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
[28] and , Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). | MR | Zbl
[29] , An analysis of traffic flow. Oper. Res. 7 (1959) 79-85. | MR
[30] , and , Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edn., Springer-Verlag, Berlin (1993). | MR | Zbl
[31] , Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48 (1995) 1305-1342. | MR | Zbl
[32] , and , On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. 29 (1976) 297-322. | MR | Zbl
[33] and , Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: M2AN 35 (2001) 239-269. | EuDML | Numdam | MR | Zbl
[34] , and , Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal. 22 (2002) 623-664. | MR | Zbl
[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 | Zbl
[36] , First order quasilinear equations in several independent space variables. Math. USSR Sb. 10 (1970) 217-243. | Zbl
[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
[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 | Zbl
[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 | Zbl
[40] , Adaptive Multiscale Schemes for Conservation Laws. Springer-Verlag, Berlin (2003). | MR | Zbl
[41] and , Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J. Sci. Comp. 30 (2007) 493-531. | MR | Zbl
[42] , Traveling-wave solutions of the diffusively corrected kinematic-wave model. Math. Comp. Modelling 35 (2002) 561-579. | MR | Zbl
[43] , Shock waves on the highway. Oper. Res. 4 (1956) 42-51. | MR
[44] and , An adaptive multiresolution method for combustion problems: Application to flame ball-vortex interaction. Comput. Fluids 34 (2005) 817-831. | Zbl
[45] , , and , A conservative fully adaptive multiresolution algorithm for parabolic conservation laws. J. Comput. Phys. 188 (2003) 493-523. | MR | Zbl
[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 | Zbl
[48] and , Numerische Mathematik 2. 3rd Edn., Springer-Verlag, Berlin (1990). | MR | Zbl
[49] and , An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003). | MR | Zbl
[50] , Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000) 681-698. | MR | Zbl
[51] , A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197-1218. | MR | Zbl
Cité par Sources :
