# Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension

Raimund Bürger; Ricardo Ruiz; Kai Schneider; Mauricio Sepúlveda

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 42, Issue: 4, page 535-563
- ISSN: 0764-583X

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topBürger, Raimund, et al. "Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension." ESAIM: Mathematical Modelling and Numerical Analysis 42.4 (2008): 535-563. <http://eudml.org/doc/250404>.

@article{Bürger2008,

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, whose
leaves are the non-uniform finite volumes on which the numerical divergence
is eventually evaluated. Moreover using the L1 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.
},

author = {Bürger, Raimund, Ruiz, Ricardo, Schneider, Kai, Sepúlveda, Mauricio},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Degenerate parabolic equation; adaptive multiresolution
scheme; monotone scheme; upwind difference scheme; boundary conditions;
entropy solution.; degenerate parabolic equation; adaptive multiresolution scheme; entropy solution; error bounds; adaptive mesh refinement},

language = {eng},

month = {5},

number = {4},

pages = {535-563},

publisher = {EDP Sciences},

title = {Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension},

url = {http://eudml.org/doc/250404},

volume = {42},

year = {2008},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2008/5//

PB - EDP Sciences

VL - 42

IS - 4

SP - 535

EP - 563

AB -
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 L1 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.

LA - eng

KW - Degenerate parabolic equation; adaptive multiresolution
scheme; monotone scheme; upwind difference scheme; boundary conditions;
entropy solution.; degenerate parabolic equation; adaptive multiresolution scheme; entropy solution; error bounds; adaptive mesh refinement

UR - http://eudml.org/doc/250404

ER -

## References

top- R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer.10 (2001) 1–102.
- J. Bell, M.J. Berger, J. Saltzman and M. Welcome, Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput.15 (1994) 127–138.
- M.J. Berger and R.J. LeVeque, Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems. SIAM J. Numer. Anal.35 (1998) 2298–2316.
- M.J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys.53 (1984) 484–512.
- S. Berres, R. Bürger, K.H. Karlsen and E.M. Tory, 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 BVt 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.
- R. Bürger, A. Coronel and M. Sepúlveda, 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.
- R. Bürger, A. Coronel and M. Sepúlveda, 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.
- 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.
- R. Bürger, R. Ruiz, K. Schneider and M. Sepúlveda, 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.
- A. Cohen, S. Kaber, S. Müller and M. Postel, Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comp.72 (2002) 183–225.
- M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comp.34 (1980) 1–21.
- P. Deuflhard and F. Bornemann, Scientific Computing with Ordinary Differential Equations. Springer-Verlag, New York (2002).
- A.C. Dick, Speed/flow relationships within an urban area. Traffic Eng. Control8 (1966) 393–396.
- M. Domingues, O. Roussel and K. Schneider, An adaptive multiresolution method for parabolic PDEs with time step control. ESAIM: Proc.16 (2007) 181–194.
- M. Domingues, S. Gomes, O. Roussel and K. Schneider, An adaptive multiresolution scheme with local time-stepping for evolutionary PDEs. J. Comput. Phys.227 (2008) 3758–3780.
- B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp.36 (1981) 321–351.
- 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.
- S. Evje and K.H. Karlsen, Monotone difference approximations of BV solutions to degenerate convection-diffusion equations. SIAM J. Numer. Anal.37 (2000) 1838–1860.
- R. Eymard, T. Gallouët, R. Herbin and A. Michel, Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math.92 (2002) 41–82.
- E. Fehlberg, Low order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. Computing6 (1970) 61–71.
- E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).
- H. Greenberg, An analysis of traffic flow. Oper. Res.7 (1959) 79–85.
- E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edn., Springer-Verlag, Berlin (1993).
- A. Harten, Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math.48 (1995) 1305–1342.
- A. Harten, J.M. Hyman and P.D. Lax, On finite-difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math.29 (1976) 297–322.
- K.H. Karlsen and N.H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: M2AN35 (2001) 239–269.
- K.H. Karlsen, N.H. Risebro and J.D. Towers, Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal.22 (2002) 623–664.
- K.H. Karlsen, N.H. Risebro and J.D. Towers, L1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vid. Selsk. (2003) 1–49.
- S.N. Kružkov, First order quasilinear equations in several independent space variables. Math. USSR Sb.10 (1970) 217–243.
- N.N. Kuznetsov, 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.
- M.J. Lighthill and G.B. Whitham, On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London Ser. A229 (1955) 317–345.
- A. Michel and J. Vovelle, 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.
- S. Müller, Adaptive Multiscale Schemes for Conservation Laws. Springer-Verlag, Berlin (2003).
- S. Müller and Y. Stiriba, Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J. Sci. Comp.30 (2007) 493–531.
- P. Nelson, Traveling-wave solutions of the diffusively corrected kinematic-wave model. Math. Comp. Modelling35 (2002) 561–579.
- P.I. Richards, Shock waves on the highway. Oper. Res.4 (1956) 42–51.
- O. Roussel and K. Schneider, An adaptive multiresolution method for combustion problems: Application to flame ball-vortex interaction. Comput. Fluids34 (2005) 817–831.
- O. Roussel, K. Schneider, A. Tsigulin and H. Bockhorn, A conservative fully adaptive multiresolution algorithm for parabolic conservation laws. J. Comput. Phys.188 (2003) 493–523.
- 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 Mathematics1697, 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).
- J.D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal.38 (2000) 681–698.
- J.D. Towers, A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal.39 (2001) 1197–1218.

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