Adaptive rational spectral methods for the linear stability analysis of nonlinear fourth-order problems

Journal of Computational Physics

Volumn 228, Issue 17, 2009, Pages 6536-6552

  Cueto Felgueroso, Luis ^a   Juanes, Ruben ^a  

^a Massachusetts Institute of Technology   (United States)

Author keywords

Adaptivity; Flow in porous media; Linear stability analysis; Phase field models; Rational approximation; Spectral methods; Thin films

Indexed keywords

EIGENVALUES AND EIGENFUNCTIONS; PHASE INTERFACES; POROUS MATERIALS; THIN FILMS;

ADAPTIVITY; DISCRETIZATIONS; FLOWS IN POROUS MEDIA; FOURTH ORDER PROBLEMS; MODELING EQUATIONS; PHASE FIELD MODELS; RATIONAL APPROXIMATIONS; SPECTRAL METHODS; THIN FILM FLOW; THIN-FILMS;

LINEAR STABILITY ANALYSIS;

EID: 67650104293     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2009.05.045     Document Type: Article

References (86)

1. Babchin A., Brailovsky I., Gordon P., and Sivashinsky G. Fingering instability in immiscible displacement. Phys. Rev. E 77 (2008) 026301
2. Baltensperger R., and Berrut J.-P. The linear rational collocation method. J. Comput. Appl. Math. 134 (2001) 243-258
3. Baltensperger R., Berrut J.-P., and Noël B. Exponential convergence of a linear rational interpolant between transformed Chebyshev points. Math. Comput. 68 (1999) 1109-1120
4. Bauters T.W.J., DiCarlo D.A., Steenhuis T.S., and Parlange J.-Y. Soil water content dependent wetting front characteristics in sands. J. Hydrol. 231 (2000) 244-254
5. Bayliss A., Class A., and Matkowsky B.J. Roundoff error in computing derivatives using the Chebyshev differentiation matrix. J. Comput. Phys. 116 (1995) 380-383
6. Bayliss A., and Matkowsky B.J. Fronts, relaxation oscillations and period doubling in solid fuel combustion. J. Comput. Phys. 71 (1987) 147-168
7. Bayliss A., and Turkel E. Mappings and accuracy for Chebyshev pseudo-spectral approximations. J. Comput. Phys. 101 (1992) 349-359
8. Bear J. Dynamics of Fluids in Porous Media (1972), Elsevier, New York
9. Berrut J.-P. Rational functions for guaranteed and experimentally well-conditioned global interpolation. Math. Comput. 15 (1988) 1-16
10. Berrut J.-P. The barycentric weights of rational interpolation with prescribed poles. J. Comput. Appl. Math. 86 (1997) 45-52
11. Berrut J.-P., and Baltensperger R. The linear rational pseudospectral method for boundary value problems. BIT 41 (2001) 868-879
12. Berrut J.-P., and Mittelmann H.D. Lebesgue constant minimizing linear rational interpolation of continuous functions over the interval. Comput. Math. Appl. 33 (1997) 77-86
13. Berrut J.-P., and Mittelmann H.D. Rational interpolation through the optimal attachment of poles to the interpolating polynomial. Numer. Algor. 23 (2000) 315-328
14. Berrut J.-P., and Mittelmann H.D. The linear rational pseudospectral method with iteratively optimized poles for two-point boundary value problems. SIAM J. Sci. Comput. 23 (2001) 961-975
15. Berrut J.-P., and Mittelmann H.D. Adaptive points shifts in rational approximation with optimized denominator. J. Comput. Appl. Math. 164 (2004) 81-92
16. Berrut J.-P., and Mittelmann H.D. Optimized points shifts and poles in the linear rational pseudospectral method for boundary value problems. J. Comput. Phys. 204 (2005) 292-301
17. Berrut J.-P., and Trefethen L.N. Barycentric Lagrange interpolation. SIAM Rev. 46 (2004) 501-517
18. Bertozzi A.L., and Brenner M.P. Linear stability and transient growth in driven contact lines. Phys. Fluids 9 (1997) 530-539
19. Boettinger W.J., Warren J.A., Beckermann C., and Karma A. Phase-field simulation of solidification. Annu. Rev. Mater. Res. 32 (2002) 163-194
20. Boyd J.P. Why Newton's method is hard for traveling waves: small denominators, KAM theory, Arnold's linear Fourier problem, non-uniqueness, constraints and erratic failure. Math. Comput. Simul. 74 (2007) 72-81
21. Cahn J.W. On spinodal decomposition. Acta Met. 9 (1961) 795-801
22. Cahn J.W., and Hilliard J.E. Free energy of non-uniform systems. I. Interfacial free energy. J. Chem. Phys. 28 (1958) 258-267
23. Chuoke R.L., van Meurs P., and van der Poel C. The instability of slow immiscible viscous liquid-liquid displacements in permeable media. Petrol. Trans. Am. Inst. Min. Eng. 216 (1959) 188-194
24. Cueto-Felgueroso L., and Juanes R. Nonlocal interface dynamics and pattern formation in gravity-driven unsaturated flow through porous media. Phys. Rev. Lett. 101 (2008) 244504
25. Cueto-Felgueroso L., and Juanes R. Linear stability analysis of a phase-field model of gravity-driven unsaturated flow in porous media. Phys. Rev. E 79 (2009) 036301
26. Davis J.M., Kataoka D.E., and Troian S.M. Transient dynamics and structure of optimal excitations in thermocapillary spreading: precursor film model. Phys. Fluids 18 (2006) 092101
27. Davis J.M., and Troian S.M. On a generalized approach to the linear stability of spatially nonuniform thin film flows. Phys. Fluids 15 (2003) 1344-1347
28. de Gennes P.G. Wetting: statics and dynamics. Rev. Mod. Phys. 57 (1985) 827-863
29. Van Deun J., Deckers K., Bultheel A., and Weideman J.A.C. Near-best fixed pole rational interpolation with applications in spectral methods. ACM Trans. Math. Software 35 (2008) Art. No. 14
30. DiCarlo D.A. Experimental measurements of saturation overshoot on infiltration. Water Resour. Res. 40 (2004) W04215
31. DiCarlo D.A. Capillary pressure overshoot as a function of imbibition flux and initial water content. Water Resour. Res. 43 (2007) W08402
32. DiCarlo D.A., and Blunt M.J. Determination of finger shape using the dynamic capillary pressure. Water Resour. Res. 36 9 (2000) 2781-2785
33. Diment G.A., and Watson K.K. Stability analysis of water movement in unsaturated porous materials 2. Numerical studies. Water Resour. Res. 19 (1983) 1002-1010
34. Diment G.A., and Watson K.K. Stability analysis of water movement in unsaturated porous materials 3. Experimental studies. Water Resour. Res. 21 (1985) 979-984
35. Diment G.A., Watson K.K., and Blennerhassett P.J. Stability analysis of water movement in unsaturated porous materials 1. Theoretical considerations. Water Resour. Res. 18 (1982) 1248-1254
36. Du X., Yao T., Stone W.D., and Hendrickx J.M.H. Stability analysis of the unsaturated water flow equation 2. Experimental verification. Water Resour. Res. 37 (2001) 1875-1881
37. Egorov A.G., Dautov R.Z., Nieber J.L., and Seshukov A.Y. Stability analysis of gravity-driven infiltrating flow. Water Resour. Res. 39 (2003) 1266
38. Eliassi M., and Glass R.J. On the continuum-scale modeling of gravity-driven fingers in unsaturated porous media: the inadequacy of the Richards equation with standard monotonic constitutive relations and hysteretic equations of state. Water Resour. Res. 37 (2001) 2019-2035
39. Eliassi M., and Glass R.J. On the porous-continuum modeling of gravity-driven fingers in unsaturated materials: Extension of the standard theory with a hold-back-pile-up effect. Water Resour. Res. 38 (2002) 1234
40. Emmerich H. Advances of and by phase-field modelling in condensed-matter physics. Adv. Phys. 57 (2008) 1-87
41. Floater M.S., and Hormann K. Barycentric rational interpolation with no poles and high rates of approximation. Numer. Math. 107 (1993) 315-331
42. Geiger S.L., and Durnford D.S. Infiltration in homogeneous sands and a mechanistic model of unstable flow. Soil Sci. Soc. Am. Proc. 64 (2000) 460-469
43. Glass R.J., Parlange J.-Y., and Steenhuis T.S. Wetting front instability 2. Experimental determination of relationships between system parameters and two-dimensional unstable flow field behaviour in initially dry porous media. Water Resour. Res. 25 (1989) 1195-1207
44. Gomba J.M., Diez J., Gratton R., and Gonzalez A.G. Stability study of a constant-volume thin film flow. Phys. Rev. E 76 (2007) 046308
45. Greengard L. Spectral integration and two-point boundary value problems. SIAM J. Numer. Anal. 28 (1991) 1071-1080
46. Greenspan H.P. On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84 (1978) 125-143
47. Guillard H., and Peyret R. On the use of spectral methods for the numerical solution of stiff problems. Comput. Methods Appl. Mech. Eng. 66 (1988) 17-43
48. E Hill D., and Parlange J.-Y. Wetting front instability in homogeneous soils. Soil Sci. Soc. Am. Proc. 36 (1972) 697-702
49. Hocking L.M. Spreading and instability of a viscous fluid sheet. J. Fluid Mech. 211 (1990) 373-392
50. Huppert H.E. Flow and instability of a viscous current down a slope. Nature 300 (1982) 427-429
51. Kapoor V. Criterion for instability of steady-state unsaturated flows. Transport Porous Med. 25 (1996) 313-334
52. Kondic L., and Bertozzi A.L. Nonlinear dynamics and transient growth of driven contact lines. Phys. Fluids 11 (1999) 3560-3562
53. Kondic L., and Diez J. Flow of thin films on patterned surfaces: controlling the instability. Phys. Rev. E 65 (2002) 045301
54. - 1) time step restriction. J. Comput. Phys. 104 (1993) 457-469
55. Lenormand R. Pattern growth and fluid displacements through porous media. Physica A 140 (1986) 114-123
56. Lenormand R., Touboul E., and Zarcone C. Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189 (1988) 165-187
57. Leverett M.C. Capillary behavior of porous solids. Petrol. Trans. Am. Inst. Min. Eng. 142 (1941) 52-169
58. Løvoll G., Méheust Y., Toussaint R., Schmittbuhl J., and Måløy K.J. Growth activity during fingering in a porous Hele-Shaw cell. Phys. Rev. E 70 (2004) 026301
59. Lu T.X., Biggar J.W., and Nielsen D.R. Water movement in glass bead porous media 2. Experiments of infiltration and finger flow. Water Resour. Res. 30 (1994) 3283-3290
60. Mualem Y. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12 3 (1976) 513-522
61. Nieber J.L., Dautov R.Z., Egorov A.G., and Sheshukov A.Y. Dynamic capillary pressure mechanism for instability in gravity-driven flows; review and extension to very dry conditions. Transport Porous Med. 58 (2005) 147-172
62. Parlange J.-Y., and Hill D.E. Theoretical analysis of wetting front instability in soils. Soil Sci. 122 (1976) 236-239
63. Philip J.R. Theory of infiltration. In: Chow V.T. (Ed). Advances in Hydroscience (1969), Academic Press, New York 215-296
64. Philip J.R. Stability analysis of infiltration. Soil Sci. Soc. Am. Proc. 39 (1975) 1042-1049
65. Riaz A., and Tchelepi H.A. Numerical simulation of immiscible two-phase flow in porous media. Phys. Fluids 18 (2006) 014104
66. Richards L.A. Capillary conduction of liquids through porous mediums. Physics 1 (1931) 318-331
67. Saffman P.G., and Taylor G.I. The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid. Proc. Roy. Soc. Lond. 245 (1958) 312-329
68. Sahimi M. Flow phenomena in rocks - from continuum models to fractals, percolation,cellular-automata, and simulated annealing. Rev. Mod. Phys. 65 4 (1993) 1393-1534
69. Selker J.S., Leclerq P., Parlange J.-Y., and Steenhuis T.S. Fingered flow in two dimensions 1. Measurement of matric potential. Water Resour. Res. 28 (1992) 2513-2521
70. Selker J.S., Parlange J.-Y., and Steenhuis T.S. Fingered flow in two dimensions 2. Predicting finger moisture profile. Water Resour. Res. 28 (1992) 2523-2528
71. Shiozawa S., and Fujimaki H. Unexpected water content profiles under flux-limited one-dimensional downward infiltration in initially dry granular media. Water Resour. Res. 40 (2004) W07404
72. Sililo O.T.N., and Tellam J.H. Fingering in unsaturated zone flow: a qualitative review with laboratory experiments in heterogeneous systems. Ground Water 38 (2000) 864-871
73. T.W. Tee, An adaptive rational spectral method for differential equations with rapidly varying solutions. Ph.D. Thesis, vol. 28, 2006, pp. 1798-1811.
74. Tee T.W., and Trefethen L.N. A rational spectral collocation method with adaptively transformed Chebyshev grid points. SIAM J. Sci. Comput. 28 (2006) 1798-1811
75. Tiwari N., Mester Z., and Davis J.M. Stability and transient dynamics of thin liquid films flowing over locally heated surfaces. Phys. Rev. E 76 (2007) 056306
76. Trefethen L.N. Spectral Methods in Matlab (2000), Society for Industrial and Applied Mathematics (SIAM)
77. Troian S.M., Herbolzheimer E., Safran S.A., and Joanny J.F. Fingering instabilities of driven spreading films. Phys. Fluids 10 (1989) 25-30
78. Ursino N. Linear stability analysis of infiltration analytical and numerical solution. Transport Porous Med. 38 (2000) 261-271
79. van Duijn C.J., Pieters G.J.M., and Raats P.A.C. Steady flows in unsaturated soils are stable. Transport Porous Med. 57 (2004) 215-244
80. van Genuchten M.T. A closed form equation to predict the water conductivity of unsaturated soil. Soil Sci. Soc. Am. Proc. 44 (1980) 892-898
81. Wang Z., Jury W.A., Tuli A., and Kim D.-J. Unstable flow during redistribution: controlling factors and practical implications. Vadose Zone J. 3 (2004) 549-559
82. Wang Z., Tuli A., and Jury W.A. Unstable flow during redistribution in homogeneous soil. Vadose Zone J. 2 (2003) 52-60
83. Weideman J.A.C. Computing the dynamics of complex singularities of nonlinear PDEs. SIAM J. Appl. Dyn. Syst. 2 (2003) 171-186
84. Weideman J.A.C., and Trefethen L.N. The eigenvalues of second-order spectral differentiation matrices. SIAM J. Numer. Anal. 25 (1988) 1279-1298
85. Weitz D.A., Stokes J.P., Ball R.C., and Kushnick A.P. Dynamic capillary pressure in porous media: origin of the viscous fingering lengthscale. Phys. Rev. Lett. 59 (1987) 2967-2970
86. Zhornitskaya L., and Bertozzi A.L. Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37 (2000) 523-555