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Fully adaptive multiresolution schemes forstrongly degenerate parabolic equations in one space dimension

Published online by Cambridge University Press:  27 May 2008

Raimund Bürger ,
Ricardo Ruiz ,
Kai Schneider  and
Mauricio Sepúlveda
Raimund Bürger
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. rburger@ing-mat.udec.cl; rruiz@ing-mat.udec.cl; mauricio@ing-mat.udec.cl
Ricardo Ruiz
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. rburger@ing-mat.udec.cl; rruiz@ing-mat.udec.cl; mauricio@ing-mat.udec.cl
Kai Schneider
Affiliation:
Centre de Mathématiques et d'Informatique, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France. kschneid@cmi.univ-mrs.fr
Mauricio Sepúlveda
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile. rburger@ing-mat.udec.cl; rruiz@ing-mat.udec.cl; mauricio@ing-mat.udec.cl
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Abstract

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, whoseleaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the L 1 contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate thecomputational efficiency together with the convergence properties.

Keywords

Degenerate parabolic equationadaptive multiresolution schememonotone schemeupwind difference schemeboundary conditionsentropy solution.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 4 , July 2008 , pp. 535 - 563
Copyright
© EDP Sciences, SMAI, 2008

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References

Becker, R. and Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1–102. CrossRef
Bell, J., Berger, M.J., Saltzman, J. and Welcome, M., Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput. 15 (1994) 127–138. CrossRef
Berger, M.J. and LeVeque, R.J., Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal. 35 (1998) 2298–2316. CrossRef
Berger, M.J. and Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53 (1984) 484–512. CrossRef
Berres, S., Bürger, R., Karlsen, K.H. and Tory, E.M., Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math. 64 (2003) 41–80.
R. Bürger and K.H. Karlsen, On some upwind schemes for the phenomenological sedimentation-consolidation model. J. Eng. Math. 41 (2001) 145–166.
R. Bürger and K.H. Karlsen, On a diffusively corrected kinematic-wave traffic model with changing road surface conditions. Math. Models Meth. Appl. Sci. 13 (2003) 1767–1799.
R. Bürger, S. Evje and K.H. Karlsen, On strongly degenerate convection-diffusion problems modeling sedimentation-consolidation processes. J. Math. Anal. Appl. 247 (2000) 517–556.
R. Bürger, K.H. Karlsen, N.H. Risebro and J.D. Towers, Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. Numer. Math. 97 (2004) 25–65.
R. Bürger, K.H. Karlsen and J.D. Towers, A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units. SIAM J. Appl. Math. 65 (2005) 882–940.
Bürger, R., Coronel, A. and Sepúlveda, M., 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. CrossRef
Bürger, R., Coronel, A. and Sepúlveda, M., 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. CrossRef
R. Bürger, A. Kozakevicius and M. Sepúlveda, Multiresolution schemes for strongly degenerate parabolic equations in one space dimension. Numer. Meth. Partial Diff. Equ. 23 (2007) 706–730. CrossRef
Bürger, R., Ruiz, R., Schneider, K. and Sepúlveda, M., Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux. J. Eng. Math. 60 (2008) 365–385.
J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rat. Mech. Anal. 147 (1999) 269–361.
G. Chiavassa, R. Donat and S. Müller, 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.
Cohen, A., Kaber, S., Müller, S. and Postel, M., Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comp. 72 (2002) 183–225. CrossRef
Crandall, M.G. and Majda, A., Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1–21. CrossRef
P. Deuflhard and F. Bornemann, Scientific Computing with Ordinary Differential Equations. Springer-Verlag, New York (2002).
Dick, A.C., Speed/flow relationships within an urban area. Traffic Eng. Control 8 (1966) 393–396.
Domingues, M., Roussel, O. and Schneider, K., An adaptive multiresolution method for parabolic PDEs with time step control. ESAIM: Proc. 16 (2007) 181–194. CrossRef
Domingues, M., Gomes, S., Roussel, O. and Schneider, K., An adaptive multiresolution scheme with local time-stepping for evolutionary PDEs. J. Comput. Phys. 227 (2008) 3758–3780. CrossRef
Engquist, B. and Osher, S., One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321–351. CrossRef
M.S. Espedal and K.H. Karlsen, 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.
Evje, S. and Karlsen, K.H., Monotone difference approximations of BV solutions to degenerate convection-diffusion equations. SIAM J. Numer. Anal. 37 (2000) 1838–1860. CrossRef
Eymard, R., Gallouët, T., Herbin, R. and Michel, A., Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92 (2002) 41–82. CrossRef
Fehlberg, E., Low order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. Computing 6 (1970) 61–71. CrossRef
E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).
Greenberg, H., An analysis of traffic flow. Oper. Res. 7 (1959) 79–85. CrossRef
E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edn., Springer-Verlag, Berlin (1993).
Harten, A., Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48 (1995) 1305–1342. CrossRef
Harten, A., Hyman, J.M. and Lax, P.D., On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. 29 (1976) 297–322. CrossRef
Karlsen, K.H. and Risebro, N.H., Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: M2AN 35 (2001) 239–269. CrossRef
Karlsen, K.H., Risebro, N.H. and Towers, J.D., Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal. 22 (2002) 623–664. CrossRef
K.H. Karlsen, N.H. Risebro and J.D. Towers, L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vid. Selsk. (2003) 1–49.
Kružkov, S.N., First order quasilinear equations in several independent space variables. Math. USSR Sb. 10 (1970) 217–243. CrossRef
Kuznetsov, N.N., 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. CrossRef
Lighthill, M.J. and Whitham, G.B., On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London Ser. A 229 (1955) 317–345. CrossRef
Michel, A. and Vovelle, J., 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. CrossRef
S. Müller, Adaptive Multiscale Schemes for Conservation Laws. Springer-Verlag, Berlin (2003).
Müller, S. and Stiriba, Y., Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J. Sci. Comp. 30 (2007) 493–531.
Nelson, P., Traveling-wave solutions of the diffusively corrected kinematic-wave model. Math. Comp. Modelling 35 (2002) 561–579. CrossRef
Richards, P.I., Shock waves on the highway. Oper. Res. 4 (1956) 42–51. CrossRef
Roussel, O. and Schneider, K., An adaptive multiresolution method for combustion problems: Application to flame ball-vortex interaction. Comput. Fluids 34 (2005) 817–831. CrossRef
Roussel, O., Schneider, K., Tsigulin, A. and Bockhorn, H., A conservative fully adaptive multiresolution algorithm for parabolic conservation laws. J. Comput. Phys. 188 (2003) 493–523. CrossRef
R. Ruiz, Métodos de Multiresolución y su Aplicación a un Problema de Ingeniería. Tesis para optar al título de Ingeniero Matemático, Universidad de Concepción, Chile (2005).
C.-W. Shu, 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.
J. Stoer and R. Bulirsch, Numerische Mathematik 2. 3rd Edn., Springer-Verlag, Berlin (1990).
E. Süli and D.F. Mayers, An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003).
Towers, J.D., Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000) 681–698. CrossRef
Towers, J.D., A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197–1218. CrossRef