×

zbMATH — the first resource for mathematics

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux. (English) Zbl 1137.65393
Summary: A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist-Osher approximation for the flux and explicit time-stepping. An adaptive multiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier-thickener model illustrate the efficiency of this method.

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
90B20 Traffic problems in operations research
PDF BibTeX Cite
Full Text: DOI arXiv
References:
[1] Harten A (1983). High-resolution schemes for hyperbolic conservation laws. J Comput Phys 49: 357–393 · Zbl 0565.65050
[2] Shu CW and Osher S (1989). Efficient implementation of essentially non-oscillatory shock-capturing schemes II. J Comput Phys 83: 32–78 · Zbl 0674.65061
[3] Shu CW (1998). Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Cockburn, B, Johnson, C, Shu, CW and Tadmor, E (eds) Advanced numerical approximation of nonlinear hyperbolic equations (Quarteroni A, Ed.), Lecture Notes in Mathematics, vol 1697, pp 325–432. Springer-Verlag, Berlin
[4] Kurganov A and Tadmor E (2000). New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. J Comput Phys 160: 241–282 · Zbl 0987.65085
[5] Nessyahu H and Tadmor E (1990). Non-oscillatory central differencing for hyperbolic conservation laws. J Comput Phys 87: 408–463 · Zbl 0697.65068
[6] Toro EF and Billett SJ (2000). Centered TVD schemes for hyperbolic conservation laws. IMA J Numer Anal 20: 47–79 · Zbl 0943.65100
[7] Harten A (1995). Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm Pure Appl Math 48: 1305–1342 · Zbl 0860.65078
[8] Bihari BL and Harten A (1995). Application of generalized wavelets: a multiresolution scheme. J Comp Appl Math 61: 275–321 · Zbl 0840.65092
[9] Roussel O, Schneider K, Tsigulin A and Bockhorn H (2003). A conservative fully adaptive multiresolution algorithm for parabolic PDEs. J Comput Phys 188: 493–523 · Zbl 1022.65093
[10] Dahmen W, Gottschlich-Müller B and Müller S (2001). Multiresolution schemes for conservation laws. Numer Math 88: 399–443 · Zbl 1001.65104
[11] Cohen A, Kaber SM, Müller S and Postel M (2001). Fully adaptive multiresolution finite-volume schemes for conservation laws. Math Comp 72: 183–225 · Zbl 1010.65035
[12] Müller S (2003). Adaptive multiscale schemes for conservation laws. Lecture Notes in Computational Science and Engineering, vol 27. Springer-Verlag, Berlin · Zbl 1016.76004
[13] Berres S, Bürger R, Karlsen KH and Tory EM (2003). Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J Appl Math 64: 41–80 · Zbl 1047.35071
[14] Bürger R, Evje S and Karlsen KH (2000). On strongly degenerate convection–diffusion problems modeling sedimentation–consolidation processes. J Math Anal Appl 247: 517–556 · Zbl 0961.35078
[15] Espedal MS and Karlsen KH (2000). Numerical solution of reservoir flow models based on large time step operator splitting methods. In: Espedal, MS, Fasano, A and Mikelić, A (eds) Filtration in porous media and industrial application, Lecture Notes in Mathematics, vol 1734, pp 9–77. Springer-Verlag, Berlin
[16] Bürger R and Karlsen KH (2003). On a diffusively corrected kinematic-wave traffic model with changing road-surface conditions. Math Models Meth Appl Sci 13: 1767–1799 · Zbl 1055.35071
[17] Nelson P (2002). Traveling-wave solutions of the diffusively corrected kinematic-wave model. Math Comp Model 35: 561–579 · Zbl 0994.90031
[18] Mochon S (1987). An analysis of the traffic on highways with changing surface conditions. Math Model 9: 1–11
[19] Bürger R, Karlsen KH, Risebro NH and Towers JD (2004). Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier–thickener units. Numer Math 97: 25–65 · Zbl 1053.76047
[20] Bürger R, Karlsen KH and Towers JD (2005). A mathematical model of continuous sedimentation of flocculated suspensions in clarifier–thickener units. SIAM J Appl Math 65: 882–940 · Zbl 1089.76061
[21] Karlsen KH, Risebro NH and Towers JD (2002). On an upwind difference scheme for degenerate parabolic convection–diffusion equations with a discontinuous coefficient. IMA J Numer Anal 22: 623–664 · Zbl 1014.65073
[22] Karlsen KH, Risebro NH and Towers JD (2003). L 1 stability for entropy solutions of nonlinear degenerate parabolic convection–diffusion equations with discontinuous coefficients. Skr K Nor Vid Selsk 3: 1–49 · Zbl 1036.35104
[23] Harten A (1994). Adaptive multiresolution schemes for shock computations. J Comput Phys 115: 319–338 · Zbl 0925.65151
[24] Harten A (1996). Multiresolution representation of data: a general framework. SIAM J Numer Anal 33: 1205–1256 · Zbl 0861.65130
[25] Jameson L (1998). A wavelet-optimized, very high order adaptive grid and order numerical method. SIAM J Sci Comput 19: 1980–2013 · Zbl 0913.65090
[26] Holmström M (1999). Solving hyperbolic PDEs using interpolating wavelets. SIAM J Sci Comput 21: 405–420 · Zbl 0959.65109
[27] Bürger R and Kozakevicius A (2007). Adaptive multiresolution WENO schemes for multi-species kinematic flow models. J Comput Phys 224: 1190–1222 · Zbl 1123.65305
[28] Verfürth R (1995). A review of a posteriori estimation and adaptative mesh-refinement techniques. Advances in numerical mathematics. Wiley/Teubner, New York-Stuttgart
[29] Eriksson K and Johnson C (1991). Adaptative finite element methods for parabolic problems. I. A linear model problem. SIAM J Numer Anal 28: 43–77 · Zbl 0732.65093
[30] Ohlberger M (2001). A posteriori error estimates for vertex centered finite-volume approximations of convection–diffusion– reaction equations. M2AN Math Model Numer Anal 35: 355–387 · Zbl 0992.65100
[31] Tadmor E (1991). Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J Numer Anal 28: 891–906 · Zbl 0732.65084
[32] Cockburn B, Coquel F and LeFloch P (1994). An error estimate for finite-volume methods for multidimensional conservation laws. Math Comp 63: 77–103 · Zbl 0855.65103
[33] Bihari BL and Harten A (1997). Multiresolution schemes for the numerical solution of 2-D conservation laws I. SIAM J Sci Comput 18: 315–354 · Zbl 0878.35007
[34] Chiavassa G and Donat R (2001). Point value multiscale algorithms for 2D compressive flows. SIAM J Sci Comput 23: 805–823 · Zbl 1043.76046
[35] Roussel O and Schneider K (2006). Numerical studies of thermodiffusive flame structures interacting with adiabatic walls using an adaptive multiresolution scheme. Combust Theory Model 10: 273–288 · Zbl 1121.80330
[36] Kružkov SN (1970). First order quasi-linear equations in several independent variables. Math USSR Sb 10: 217–243 · Zbl 0215.16203
[37] Vol’pert AI (1967). The spaces BV and quasilinear equations. Math USSR Sb 2: 225–267 · Zbl 0168.07402
[38] Vol’pert AI and Hudjaev SI (1969). Cauchy’s problem for degenerate second order quasilinear parabolic equations. Math USSR Sb 7: 365–387 · Zbl 0191.11603
[39] Carrillo J (1999). Entropy solutions for nonlinear degenerate problems. Arch Rat Mech Anal 147: 269–361 · Zbl 0935.35056
[40] Mascia C, Porretta A and Terracina A (2002). Nonhomogeneous Dirichlet problems for degenerate parabolic–hyperbolic equations. Arch Rat Mech Anal 163: 87–124 · Zbl 1027.35081
[41] Chen GQ and DiBenedetto E (2001). Stability of entropy solutions to the Cauchy problem for a class of nonlinear hyperbolic-parabolic equations. SIAM J Math Anal 33: 751–762 · Zbl 1027.35080
[42] Chen GQ and Perthame B (2003). Well-posedness for non-isotropic degenerate parabolic–hyperbolic equations. Ann Inst H Poincaré Anal Non Linéaire 20: 645–668 · Zbl 1031.35077
[43] Karlsen KH and Risebro NH (2003). On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete Contin Dyn Syst 9: 1081–1104 · Zbl 1027.35057
[44] Michel A and Vovelle J (2003). Entropy formulation for parabolic degenerate equations with general Dirichlet boundary conditions and application to the convergence of FV methods. SIAM J Numer Anal 41: 2262–2293 · Zbl 1058.35127
[45] Evje S and Karlsen KH (2000). Monotone difference approximation of BV solutions to degenerate convection–diffusion equations. SIAM J Numer Anal 37: 1838–1860 · Zbl 0985.65100
[46] Crandall MG and Majda A (1980). Monotone difference approximations for scalar conservation laws. Math Comp 34: 1–21 · Zbl 0423.65052
[47] Karlsen KH and Risebro NH (2001). Convergence of finite difference schemes for viscous and inviscid conservation laws with rough coefficients. M2AN Math Model Numer Anal 35: 239–269 · Zbl 1032.76048
[48] Bürger R, Coronel A and Sepúlveda M (2006). 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: 91–112 · Zbl 1082.65081
[49] Engquist B and Osher S (1981). One-sided difference approximations for nonlinear conservation laws. Math Comp 36: 321–351 · Zbl 0469.65067
[50] Bürger R, Karlsen KH, Klingenberg C and Risebro NH (2003). A front tracking approach to a model of continuous sedimentation in ideal clarifier–thickener units. Nonlin Anal Real World Appl 4: 457–481 · Zbl 1013.35052
[51] Bürger R, Karlsen KH and Risebro NH (2005). A relaxation scheme for continuous sedimentation in ideal clarifier–thickener units. Comput Math Appl 50: 993–1009 · Zbl 1122.76063
[52] Karlsen KH, Klingenberg C and Risebro NH (2003). A relaxation scheme for conservation laws with a discontinuous coefficient. Math Comp 73: 1235–1259 · Zbl 1078.65076
[53] Karlsen KH, Risebro NH and Towers JD (2002). On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient. Electron J Diff Eqns 93: 1–23
[54] Bürger R, Kozakevicius A and Sepúlveda M (2007). Multiresolution schemes for degenerate parabolic equations in one space dimension. Numer Meth Partial Diff Eqns 23: 706–730 · Zbl 1114.65120
[55] Bürger R, Ruiz R, Schneider K, Sepúlveda M (2007) Fully adaptive multiresolution schemes for strongly degenerate parabolic equations in one space dimension. Preprint 2007-03, Departamento de Ingeniería Matemática, Universidad de Concepción (submitted) · Zbl 1147.65066
[56] Lighthill MJ and Whitham GB (1955). On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc Roy Soc London Ser A 229: 317–345 · Zbl 0064.20906
[57] Richards PI (1956). Shock waves on the highway. Oper Res 4: 42–51
[58] Dick AC (1966). Speed/flow relationships within an urban area. Traffic Eng Control 8: 393–396
[59] Greenberg H (1959). An analysis of traffic flow. Oper Res 7: 79–85
[60] Kuznetsov NN (1976). Accuracy of some approximate methods for computing the weak solutions of a first order quasilinear equation. USSR Comp Math and Math Phys 16: 105–119 · Zbl 0381.35015
[61] Bürger R and Karlsen KH (2001). On some upwind schemes for the phenomenological sedimentation-consolidation model. J Eng Math 41: 145–166 · Zbl 1128.76341
[62] Helbing D (1997). Verkehrsdynamik. Springer-Verlag, Berlin
[63] Kerner BS (2004). The physics of traffic. Springer-Verlag, Berlin
[64] Garavello M and Piccoli B (2006). Traffic flow on networks. American Institute of Mathematical Sciences, Springfield, MO, USA · Zbl 1136.90012
[65] Bellomo N and Coscia V (2005). First order models and closure of the mass conservation equation in the mathematical theory of vehicular traffic flow. C R Mecanique 333: 843–851 · Zbl 1177.90076
[66] Bellomo N, Delitalia M and Coscia V (2002). On the mathematical theory of vehicular traffic flow I. Fluid dynamic and kinetic modelling. Math Models Meth Appl Sci 12: 1801–1843 · Zbl 1041.76061
[67] Bellomo N, Marasco A and Romano A (2002). From the modelling of driver’s behavior to hydrodynamic models and problems of traffic flow. Nonlin Anal Real World Appl 3: 339–363 · Zbl 1005.90016
[68] Bürger R, Narváez A (2007) Steady-state, control, and capacity calculations for flocculated suspensions in clarifier–thickeners. Int J Mineral Process, (to appear)
[69] Coronel A, James F and Sepúlveda M (2003). Numerical identification of parameters for a model of sedimentation processes. Inverse Problems 19: 951–972 · Zbl 1041.35079
[70] Berres S, Bürger R, Coronel A and Sepúlveda M (2005). Numerical identification of parameters for a strongly degenerate convection–diffusion problem modelling centrifugation of flocculated suspensions. Appl Numer Math 52: 311–337 · Zbl 1062.65099
[71] Stiriba Y and Müller S (2007). Fully adaptive multiscale schemes for conservation laws employing locally varying time stepping. J Sci Comput 30: 493–531 · Zbl 1110.76037
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.