Free Access
Issue |
ESAIM: M2AN
Volume 42, Number 4, July-August 2008
|
|
---|---|---|
Page(s) | 535 - 563 | |
DOI | https://doi.org/10.1051/m2an:2008016 | |
Published online | 27 May 2008 |
- R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1–102. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- M.J. Berger and J. Oliger, Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53 (1984) 484–512. [NASA ADS] [CrossRef] [MathSciNet] [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- R. Bürger and K.H. Karlsen, On some upwind schemes for the phenomenological sedimentation-consolidation model. J. Eng. Math. 41 (2001) 145–166. [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- 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. [MathSciNet] [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- 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. [Google Scholar]
- J. Carrillo, Entropy solutions for nonlinear degenerate problems. Arch. Rat. Mech. Anal. 147 (1999) 269–361. [Google Scholar]
- 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. [Google Scholar]
- A. Cohen, S. Kaber, S. Müller and M. Postel, Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comp. 72 (2002) 183–225. [Google Scholar]
- M.G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws. Math. Comp. 34 (1980) 1–21. [Google Scholar]
- P. Deuflhard and F. Bornemann, Scientific Computing with Ordinary Differential Equations. Springer-Verlag, New York (2002). [Google Scholar]
- A.C. Dick, Speed/flow relationships within an urban area. Traffic Eng. Control 8 (1966) 393–396. [Google Scholar]
- M. Domingues, O. Roussel and K. Schneider, An adaptive multiresolution method for parabolic PDEs with time step control. ESAIM: Proc. 16 (2007) 181–194. [CrossRef] [EDP Sciences] [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- B. Engquist and S. Osher, One-sided difference approximations for nonlinear conservation laws. Math. Comp. 36 (1981) 321–351. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- E. Fehlberg, Low order classical Runge-Kutta formulas with step size control and their application to some heat transfer problems. Computing 6 (1970) 61–71. [CrossRef] [Google Scholar]
- E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). [Google Scholar]
- H. Greenberg, An analysis of traffic flow. Oper. Res. 7 (1959) 79–85. [CrossRef] [Google Scholar]
- E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd Edn., Springer-Verlag, Berlin (1993). [Google Scholar]
- A. Harten, Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48 (1995) 1305–1342. [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- K.H. Karlsen and N.H. Risebro, Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. ESAIM: M2AN 35 (2001) 239–269. [CrossRef] [EDP Sciences] [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- 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. [Google Scholar]
- S.N. Kružkov, First order quasilinear equations in several independent space variables. Math. USSR Sb. 10 (1970) 217–243. [Google Scholar]
- 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. [Google Scholar]
- 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. A 229 (1955) 317–345. [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- S. Müller, Adaptive Multiscale Schemes for Conservation Laws. Springer-Verlag, Berlin (2003). [Google Scholar]
- S. Müller and Y. Stiriba, Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J. Sci. Comp. 30 (2007) 493–531. [Google Scholar]
- P. Nelson, Traveling-wave solutions of the diffusively corrected kinematic-wave model. Math. Comp. Modelling 35 (2002) 561–579. [CrossRef] [Google Scholar]
- P.I. Richards, Shock waves on the highway. Oper. Res. 4 (1956) 42–51. [Google Scholar]
- O. Roussel and K. Schneider, An adaptive multiresolution method for combustion problems: Application to flame ball-vortex interaction. Comput. Fluids 34 (2005) 817–831. [CrossRef] [Google Scholar]
- 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. [CrossRef] [MathSciNet] [Google Scholar]
- 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). [Google Scholar]
- 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. [Google Scholar]
- J. Stoer and R. Bulirsch, Numerische Mathematik 2. 3rd Edn., Springer-Verlag, Berlin (1990). [Google Scholar]
- E. Süli and D.F. Mayers, An Introduction to Numerical Analysis. Cambridge University Press, Cambridge (2003). [Google Scholar]
- J.D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38 (2000) 681–698. [Google Scholar]
- J.D. Towers, A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39 (2001) 1197–1218. [CrossRef] [MathSciNet] [Google Scholar]
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.