Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-11T03:10:27.035Z Has data issue: false hasContentIssue false
  • English
  • Français

Velocity-Based Moving Mesh Methods for Nonlinear Partial Differential Equations

Published online by Cambridge University Press:  20 August 2015

M. J. Baines ,
M. E. Hubbard  and
P. K. Jimack
M. J. Baines*
Affiliation:
Department of Mathematics, The University of Reading, RG6 6AX, UK
M. E. Hubbard*
Affiliation:
School of Computing, University of Leeds, LS2 9JT, UK
P. K. Jimack*
Affiliation:
School of Computing, University of Leeds, LS2 9JT, UK
*
Corresponding author.Email:m.j.baines@reading.ac.uk
Email:m.e.hubbard@leeds.ac.uk
Email:p.k.jimack@leeds.ac.uk

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This article describes a number of velocity-based moving mesh numerical methods for multidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.

Keywords

35R3565M6076M10Time-dependent nonlinear diffusionmoving boundariesfinite element methodLagrangian meshesconservation of mass
Type
Review Article
Information
Communications in Computational Physics , Volume 10 , Issue 3 , September 2011 , pp. 509 - 576
Copyright
Copyright © Global Science Press Limited 2011

References

[1]Abgrall, R., Residual distribution schemes: current status and future trends, Comput. Fluids, 214 (2006), 773–808.Google Scholar
[2]Ahamadi, M. and Harlen, O. G., A Lagrangian finite element method for simulation of a suspension under planar extensional flow, J. Comput. Phys., 227 (2008), 7543–7560.Google Scholar
[3]Aronson, D., The porous medium equation, in Nonlinear Diffusion Problems, Lect. Notes Math., 1224 (1986), 1–46.Google Scholar
[4]Baer, T. A., Cairncross, R. A., Schunk, P. R., Rao, R. R. and Sackinger, P. A., A finite element method for free surface flows of incompressible fluids in three dimensions, part II: dynamic wetting lines, Int. J. Numer. Meth. Fluids, 33 (2000), 405–427.Google Scholar
[5]Baines, M. J., Moving Finite Elements, Oxford University Press, 1994.Google Scholar
[6]Baines, M. J., Grid adaptation via node movement, Appl. Numer. Math., 26 (1998), 77–96.Google Scholar
[7]Baines, M. J., Hubbard, M. E. and Jimack, P. K., A moving mesh finite element algorithm for the adaptive solution of time-dependent partial differential equations with moving boundaries, Appl. Numer. Math., 54 (2005), 450–469.Google Scholar
[8]Baines, M. J., Hubbard, M. E. and Jimack, P. K., A moving mesh finite element algorithm for fluid flow problems with moving boundaries, Int. J. Numer. Meth. Fluids, 47(10/11) (2005), 1077–1083.Google Scholar
[9]Baines, M. J., Hubbard, M. E., Jimack, P. K. and Jones, A. C., Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions, Appl. Numer. Math., 56 (2006), 230–252.Google Scholar
[10]Baines, M. J., Hubbard, M. E., Jimack, P. K. and Mahmood, R., A moving-mesh finite element method and its application to the numerical solution of phase-change problems, Commun. Comput. Phys., 6(3) (2009), 595–624.Google Scholar
[11]Baines, M. J. and Wathen, A. J., Moving finite element modelling of compressible flow, Appl. Numer. Math., 2 (1986), 495–514.Google Scholar
[12]Baines, M. J. and Wathen, A. J., Moving finite elements for evolutionary problems, I, theory, J. Comput. Phys., 79 (1988), 245–269.Google Scholar
[13]Balmforth, N. J., Burbidge, A. S., Craster, R. V., Salzig, J. and Shen, A., Visco-plastic models of isothermal lava domes, J. Fluid Mech., 403 (2000), 37–65.Google Scholar
[14]Barenblatt, G. I., Scaling, Self-Similarity and Intermediate Asymptotics, Cambridge University Press, 1996.Google Scholar
[15]Barenblatt, G. I., Scaling, Cambridge University Press, 2003.Google Scholar
[16]Barrett, J. W., Langdon, S. and Nurnberg, R., Finite element approximation of a sixth-order nonlinear degenerate parabolic equation, Numer. Math., 96 (2004), 401–434.Google Scholar
[17]Barlow, A., A compatible finite element multi-material ALE hydrodynamics algorithm, Int. J. Numer. Meth. Fluids, 56 (2008), 953–964.Google Scholar
[18]Berger, A. E., Ciment, M. and Rogers, J. C. W., Numerical solution of a diffusion consumption problem with a free boundary, SIAM J. Numer. Anal., 12 (1975), 646–672.Google Scholar
[19]Bhattacharya, B., A Moving Finite Element Method for High Order Nonlinear Diffusion Problems, MSc dissertation, Department of Mathematics, University of Reading, UK, 2006.Google Scholar
[20]Blake, K. W., Moving Mesh Methods for Nonlinear Partial Differential Equations, PhD thesis, Department of Mathematics, University of Reading, UK, 2001.Google Scholar
[21]Blake, K. W. and Baines, M. J., Moving mesh methods for nonlinear partial differential equations, Numerical Analysis Report 7/01, Dept of Mathematics, University of Reading, UK, 2001.Google Scholar
[22]Blowey, J. F., King, J. R. and Langdon, S., Small and waiting time behaviour of the thin film equation, SIAM J. Appl. Math., 67 (2007), 1776–1807.CrossRefGoogle Scholar
[23]Bochev, P., Liao, G. and de la Pena, G., Analysis and computation of adaptive moving grids by deformation, Numer. Meth. Part. Diff. Equations, 12 (1998), 489–506.Google Scholar
[24]Boffi, D. and Gastaldi, L., Stability and geometric conservation laws for ALE formulations, Comput. Method. Appl. Math., 193 (2004), 4717–4739.Google Scholar
[25]Bonacina, C., Comini, G., Fasano, A. and Primicerio, M., Numerical solution of phase-change problems, Int. J. Heat Mass Trans., 16 (1973), 1825–1832.CrossRefGoogle Scholar
[26]de Boor, C., Good approximation by splines with variable knots II, Springer Lecture Notes Series, 363 (1973).Google Scholar
[27]de Berg, M., Cheong, O., Kreveld, M. van and Overmars, M., Computational Geometry: Algorithms and Applications, Springer-Verlag, 2008.Google Scholar
[28]Budd, C. J. and Piggott, M. D., The geometric integration of scale invariant ordinary and partial differential equations, J. Comput. Appl. Math., 128 (2001), 399–422.Google Scholar
[29]Budd, C. J., Huang, W. and Russell, R. D., Moving mesh methods for problems with blow-up, SIAM J. Sci. Comput., 17 (1996), 305–327.Google Scholar
[30]Budd, C. J., Huang, W. and Russell, R. D., Adaptivity with moving grids, Acta. Numer., 18 (2009), 111–241.Google Scholar
[31]Budd, C., Collins, G., Huang, W. and Russell, R. D., Self-similar numerical solutions of the porous medium equation using moving mesh methods, Phil. Trans. Roy. Soc., 357 (1999), 1047–1078.Google Scholar
[32]Cai, X.-X., Jiang, B. and Liao, G., Adaptive grid generation based on least-squares finite-element method, Comput. Math. Appl., 48 (2004), 1077–1085.Google Scholar
[33]Cairncross, R. A., Schunk, P. R., Baer, T. A., Rao, R. R. and Sackinger, P. A., A finite element method for free surfaceflows of incompressible fluids in three dimensions, partI,boundary fitted mesh motion, Int. J. Numer. Meth. Fluids, 33 (2000), 375–403.Google Scholar
[34]Cao, W. M., Huang, W. Z. and Russell, R. D., A moving-mesh method based on the geometric conservation law, SIAM J. Sci. Comput., 24 (2002), 118–142.Google Scholar
[35]Cao, W. M., Huang, W. Z. and Russell, R. D., Approaches for generating moving adaptive meshes: location versus velocity, Appl. Numer. Math., 47(2) (2003), 121–138.CrossRefGoogle Scholar
[36]Caramana, E. J. and Shashkov, M. J., Elimination of artificial grid distortion and hour-glasstype motions by means of Lagrangian subzonal masses and pressures, J. Comput. Phys., 142 (1998), 521–561.CrossRefGoogle Scholar
[37]Carlson, N. N. and Miller, K., Design and application of a gradient-weighted moving finite element code I: in one dimension, SIAM J. Sci. Comput., 19 (1998), 728–765.Google Scholar
[38]Carlson, N. N. and Miller, K., Design and application of a gradient-weighted moving finite element code II: in two dimensions, SIAM J. Sci. Comput., 19 (1998), 766–798.Google Scholar
[39]Coimbra, M. do Carmo, Serano, C. and Rodrigues, A., A moving finite element method for the solution of two-dimensional time-dependent models, Appl. Numer. Math., 44 (2003), 449–469.Google Scholar
[40]Coimbra, M. do Carmo, Serano, C. and Rodrigues, A., Moving finite element method: applications to science and engineering problems, Comput. Chem. Eng., 28 (2004), 597–603.Google Scholar
[41]Chu, M.-Y., Chen, H.-M., Hsieh, C.-Y., Lin, T.-H., Hsiao, H.-Y., Liao, G. and Peng, Q., Adaptive grid generation based non-rigid image registration using mutual information for breast MRI, J. Sign. Process. Syst., 54 (2009), 45–63.Google Scholar
[42]Veen, C. J. van der, Fundamentals of Glacier Dynamics, Balkema, A. A , 1999.Google Scholar
[43]Cole, S. L., Blow-Up in a Chemotaxis Model Using a Moving Mesh Method, MSc dissertation, Department of Mathematics, University of Reading, UK, 2009.Google Scholar
[44]Cote, J., Gravel, S. and Staniforth, A., A generalized family of schemes that eliminate the spurious resonant response of semi-Lagrangian schemes to orographic forcing, Mon. Weather Rev., 123 (1995), 3605–3613.Google Scholar
[45]Cote, J., Roch, M., Staniforth, A. and Fillion, L., A variable-resolution semi-Lagrangian finite-element global-model of the shallow-water equations, Mon. Weather Rev., 121 (1993), 231–243.Google Scholar
[46]Cox, C. L., Jones, W. F., Quisenberry, V. L. and Yo, F., One-dimensional infiltration with moving finite elements and improved soil water diffusivity, Water Resour. Res., 30 (1994), 1431–1438.Google Scholar
[47]Crank, J., Free and Moving Boundary Problems, Oxford University Press, 1984.Google Scholar
[48]Crank, J. and Gupta, R. S., A moving boundary problem arising from the diffusion of oxygen in absorbing tissue, J. Int. Math. Appl., 10 (1972), 19–33.Google Scholar
[49]Dacorogna, B. and Moser, J., On a PDE involving the Jacobian determinant, Ann. I. H. Poincaré C, 7 (1990), 1–26.Google Scholar
[50]Deconinck, H. and Ricchiuto, M., Residual distribution schemes: foundation and analysis, in Encyclopedia of Computational Mechanics, Volume 3: Fluids, Stein, E., Borst, E. de, Hughes, T. J. (Eds.), John Wiley and Sons, Ltd., 2007.Google Scholar
[51]Delzanno, G. L., Chacón, L., Finn, J. M., Chung, Y. and Lapenta, G., An optimal robust equidis-tribution method for two-dimensional grid adaptation based on Monge-Kantorovich optimization, J. Comput. Phys., 227 (2008), 9841–9864.Google Scholar
[52]Demirdžić, I. and Perić, M., Space conservation law in finite volume calculations of fluid flow, Int. J. Numer. Meth. Fluids, 8(9) (1988), 1037–1050.Google Scholar
[53]Demirdžić, I. and Perić, M., Finite volume method for prediction of fluid flow in arbitrarily shaped domains with moving boundaries, Int. J. Numer. Meth. Fluids, 10(7) (1990), 771–790.Google Scholar
[54]Duarte, B., Moving finite elements method applied to the solution of front reaction models: causticizing reaction, Comput. Chem. Eng., 19 (1995), S421–S426.Google Scholar
[55]Duarte, B. P. M. and Baptista, C. M. S. G., Moving finite elements method applied to dynamic population balance equations, AIChE J., 54 (2008), 673–692.Google Scholar
[56]Etienne, S., Garon, A. and Pelletier, D., Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow, J. Comput. Phys., 228(7) (2009), 2313–2333.Google Scholar
[57]Farhat, C., Geuzaine, P. and Grandmont, C., The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of flow problems on moving grids, J. Comput. Phys., 174(2) (2001), 669–694.Google Scholar
[58]Formaggia, L. and Nobile, F., Stability analysis of second-order time accurate schemes for ALE-FEM, Comput. Method. Appl. Math., 134 (1996), 71–90.Google Scholar
[59]Garabedian, P., Partial Differential Equations, Wiley, New York, 1964.Google Scholar
[60]Gardner, W. and Widtsoe, J. A., The movement of soil moisture, Soil Sci., 11 (1921), 215–232.Google Scholar
[61]Gaskell, P. H., Jimack, P. K., Sellier, M., Thompson, H. M. and Wilson, M. C. T., Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography, J. Fluid Mech., 509 (2004), 253–280.Google Scholar
[62]Gelinas, R. J., Doss, S. K. and Miller, K., The moving finite element method: applications to general partial differential equations with multiple large gradients, J. Comput. Phys., 40 (1981), 202–249.Google Scholar
[63]Geuzaine, P., Grandmont, C. and Farhat, C., Design and analysis of ALE schemes with provable second-order time-accuracy for inviscid and viscous flow simulations, J. Comput. Phys., 191(1) (2003), 206–227.Google Scholar
[64]Giraldo, F. X., The Lagrange-Galerkin method for the two-dimensional shallow water equations on adaptive grids, Int. J. Numer. Meth. Fluids, 33(6) (2000), 789–832.Google Scholar
[65]Gottardi, G. and Venutelli, M., Moving finite element model for one-dimensional infiltration in unsaturated soil, Water Resour. Res., 28 (1992), 3259–3267.CrossRefGoogle Scholar
[66]Gottardi, G. and Venutelli, M., One-dimensional moving finite element model of solute transport, Ground Water, 32 (1994), 645–649.Google Scholar
[67]Grajewski, M., Köster, M. and Turek, S., Mathematical and numerical analysis of a robust and efficient grid deformation method in the finite element context, SIAM J. Sci. Comput., 31(2) (2009), 1539–1557.Google Scholar
[68]Griffiths, D. F., Introduction to Electrodynamics, Prentice-Hall, New Jersey, 1999.Google Scholar
[69]Guillard, H. and Farhat, C., On the significance of the geometric conservation law for flow computations on moving meshes, Comput. Method. Appl. Math., 190 (2000), 1467–1482.Google Scholar
[70]Harlen, O. G., Rallinson, J. M. and Szabo, P., A split Lagrangian-Eulerian method for simulating transient viscoelastic flows, J. Non-Newton. Fluid, 60 (1995), 81–104.Google Scholar
[71]Heil, M., An efficient solver for the fully coupled solution of large-displacement fluid-structure interaction problems, Comput. Method. Appl. Math., 193 (2004), 1–23.Google Scholar
[72]Herbst, B. M., Schoombie, S. W. and Mitchell, A. R., A moving Petrov-Galerkin method for transport equations, Int. J. Numer. Meth. Eng., 18 (1982), 1321–1336.Google Scholar
[73]Howison, S. D., Mayers, D. F. and Smith, W. R., Numerical and asymptotic solution of a sixth-order nonlinear diffusion equation and related coupled systems, IMA J. Appl. Math., 57 (1996), 79–98.Google Scholar
[74]Huang, W., Variational mesh adaptation: isotropy and equidistribution, J. Comput. Phys., 174 (2001), 903–924.Google Scholar
[75]Hubbard, M. E., Baines, M. J. and Jimack, P. K., Consistent Dirichlet boundary conditions for moving boundary problems, Appl. Numer. Math., 59(6) (2009), 1337–1353.Google Scholar
[76]Jimack, P. K., A best approximation property of the moving finite element method, SIAM J. Numer. Anal., 33 (1996), 2206–2232.Google Scholar
[77]Jimack, P. K., Optimal eigenvalue and asymptotic large time approximations using the moving finite element method, IMA J. Numer. Anal., 16 (1996), 381–398.Google Scholar
[78]Jimack, P. K., An optimal finite element mesh for elastostatic structural analysis problems, Comput. Struct., 64 (1997), 197–208.Google Scholar
[79]Jimack, P. K. and Wathen, A. J., Temporal derivatives in the finite-element method on continuously deforming grids, SIAM J. Numer. Anal., 28 (1991), 990–1003.Google Scholar
[80]Johnson, I. W. and Jayawardena, A. W., Efficient numerical solution of the dispersion equation using moving finite elements, Finite Elem. Anal. Des., 28 (1998), 241–253.Google Scholar
[81]Johnson, I. W., Baines, M. J. and Wathen, A. J., Moving finite elements for evolutionary problems, II, applications, J. Comput. Phys., 79 (1988), 270–297.Google Scholar
[82]Kistler, S. F. and Scriven, L. E., Coating flow theory by element and asymptotic analysis of the Navier-Stokes system, Int. J. Numer. Meth. Fluids, 4 (1984), 207–229.Google Scholar
[83]Kuhl, E., Hulshoff, S. and Borst, R. de, An arbitrary Lagrangian Eulerian finite-element approach for fluid-structure interaction phenomena, Int. J. Numer. Meth. Eng., 57 (2003), 117–142.Google Scholar
[84]Kuiken, H. K., Viscous sintering: the surface-tension-driven flow of a liquid form under the influence of curvature gradients at its surface, J. Fluid Mech., 214 (1990), 503–515.Google Scholar
[85]Lacey, A., Ockendon, J. and Tayler, A., Waiting-time solutions of a nonlinear diffusion equation, SIAM J. Appl. Math., 42 (1982), 1252–1264.Google Scholar
[86]Liao, G. and Anderson, D., A new approach to grid generation, Appl. Anal., 44 (1992), 285–298.Google Scholar
[87]Liao, G., Lei, Z. and Pena, G. C. de la , Adaptive grids for resolution enhancement, Shock Waves, 12 (2002), 153–156.Google Scholar
[88]Liao, G., Pan, T.-W. and Su, J., A numerical grid generator based on Moser’s deformation method, Numer. Meth. Part. Diff. Equations, 10 (1994), 21–31.Google Scholar
[89]Liao, G. and Su, J., A direct method in Dacorogna-Moser’s approach of grid generation, Appl. Anal., 49 (1993), 73–84.Google Scholar
[90]Liao, G. and Su, J., A moving grid method for (1+1) dimension, Appl. Math. Lett., 8 (1995), 47–49.Google Scholar
[91]Liao, G. and Xue, J., Moving meshes by the deformation method, J. Comput. Appl. Math., 195 (2006), 83–92.Google Scholar
[92]Liu, F., Ji, S. and Liao, G., An adaptive grid method with cell-volume control and its application to Euler flow calculations, SIAM J. Sci. Comput., 20(3) (1998), 811–825.Google Scholar
[93]Marcum, D. L. and Weatherill, N. P., Unstructured grid generation using iterative point insertion and local reconnection, AIAA J., 33 (1995) 1619–1625.Google Scholar
[94]Martinez-Herrera, J. I. and Derby, J. J., Analysis of capillary-driven viscous flows during the sintering of ceramic powders, AIChE J., 40 (1984), 1794–1803.Google Scholar
[95]Mashayek, F. and Ashgriz, N., A spine-flux method for simulating free surface flows, J. Comput. Phys., 122 (1995), 367–399.Google Scholar
[96]Mavriplis, D. J. and Yang, Z., Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes, J. Comput. Phys., 213(2) (2006), 557–573.Google Scholar
[97]Mendes, P. A. and Branco, F. A., Analysis of fluid-structure interaction by an arbitrary Lagrangian-Eulerianfinite element formulation, Int. J. Numer. Meth. Fluids, 30 (1999), 897–919.Google Scholar
[98]Mercer, J. W. and Cohen, R. M., A review of immiscible fluids in the subsurface: properties, models, characterization and remediation, J. Contaminant Hydrogeology, 6 (1990), 107–163.Google Scholar
[99]Michler, C., De Sterck, H. and Deconinck, H., An arbitrary Lagrangian Eulerian formulation for residual distribution schemes on moving grids, Comput. Fluids, 32 (2003), 59–71.Google Scholar
[100]Miller, K., Moving finite elements, II, SIAM J. Numer. Anal., 18 (1981), 1033–1057.Google Scholar
[101]Miller, K., A geometrical-mechanical interpretation of gradient-weighted moving finite elements, SIAM J. Numer. Anal., 34 (1997), 67–90.Google Scholar
[102]Miller, K., Stabilized moving finite elements for convection dominated problems, J. Sci. Comput., 24 (2005), 163–182.Google Scholar
[103]Miller, K. and Baines, M. J., Least squares moving finite elements, IMA J. Numer. Anal., 21 (2001), 621–642.Google Scholar
[104]Miller, K. and Miller, R. N., Moving finite elements, I, SIAM J. Numer. Anal., 18 (1981), 1019–1032.Google Scholar
[105]Mackenzie, J. A. and Robertson, M. L., The numerical solution of one-dimensional phase change problems using an adaptive moving mesh method, J. Comput. Phys., 161 (2000), 537–557.Google Scholar
[106]Morrison, G., Numerical Modelling of Tidal Bores Using a Moving Mesh, MSc dissertation, Department of Mathematics, University of Reading, UK, 2008.Google Scholar
[107]Moser, J., Volume elements of a Riemann manifold, T. Am. Math. Soc., 120 (1965), 286–294.Google Scholar
[108]Muttin, F., Coupez, T., Bellet, M. and Chenot, J. L., Lagrangian finite-element analysis of time-dependent free-surface flow using an automatic remeshing technique-application to metal casting, Int. J. Numer. Meth. Eng., 36 (1993), 2001–2015.Google Scholar
[109]Ockendon, J.R., The role of the Crank-Gupta model in the theory of free and moving boundary problems, Adv. Comput. Math., 6 (1996), 281–293.Google Scholar
[110]Osman, K., Numerical Schemes for a Non-Linear Diffusion Problem, MSc dissertation, Department of Mathematics, University of Reading, UK, 2005.Google Scholar
[111]Parrinello, J., Modelling Water Uptake in Rice Using Moving Meshes, MSc dissertation, Department of Mathematics, University of Reading, UK, 2008.Google Scholar
[112]Partridge, D. and Baines, M. J., A moving mesh approach to an ice sheet model, Comput. Fluids, 46 (2011), 381–386.Google Scholar
[113]Pattle, R. E., Diffusion from an instantaneous point source with a concentration-dependent coefficient, Q. J. Mech. Appl. Math., 12 (1959), 407–409.CrossRefGoogle Scholar
[114]Persson, P.-O., Bonet, J. and Peraire, J., Discontinuous Galerkin solution of the Navier-Stokes equations on deformable domains, Comput. Method. Appl. Math., 198 (2009), 1585–1595.Google Scholar
[115]Peterson, R. C., Jimack, P. K. and Kelmanson, M. A., The solution of two-dimensional free-surface problems using automatic mesh generation, Int. J. Numer. Meth. Fluids, 31 (1999), 937–960.Google Scholar
[116]Piggott, M. D., Gorman, G. J., Pain, C. C., Allison, P. A., Candy, A. S., Martin, B. T. and Wells, M. R., A new computational framework for multi-scale ocean modelling based on adapting unstructured meshes, Int. J. Numer. Meth. Fluids, 56 (2008), 1003–1015.Google Scholar
[117]Please, C. P. and Sweby, P. K., A transformation to assist numerical solution of diffusion equations, Numerical Analysis Report 5/86, Department of Mathematics, University of Reading, UK, 1986.Google Scholar
[118]Ramaswamy, B. and Kawahara, M., Lagrangian finite-element analysis applied to viscous free-surface fluid flow, Int. J. Numer. Meth. Fluids, 7 (1987), 953–984.Google Scholar
[119]Roberts, R., Modelling Glacier Flow, MSc dissertation, Department of Mathematics, University of Reading, UK, 2007.Google Scholar
[120]Robertson, N., A Moving Lagrangian Mesh Modelof a Lava Dome Volcano and Talus Slope, MSc dissertation, Department of Mathematics, University of Reading, UK, 2006.Google Scholar
[121]Saito, H. and Scriven, L. E., Study of coating flow by the finite element method, J. Comput. Phys., 42 (1981), 53–76.Google Scholar
[122]Saksono, P. H., Dettmer, W. G. and Peric, D., An adaptive remeshing strategy for fluid flows with moving boundaries and fluid-structure interaction, Int. J. Numer. Meth. Eng., 71 (2007), 1009–1050.Google Scholar
[123]Schmidt, A., Computation of three dimensional dendrites with finite elements, J. Comput. Phys., 125 (1996), 293–312.Google Scholar
[124]Semper, B. and Liao, G., A moving grid finite element method using grid deformation, Numer. Meth. Part. Diff. Equations, 11 (1995), 603–615.Google Scholar
[125]Serano, C., Rodrigues, A. and Villadsen, J., The moving finite element method with polynomial approximation of any degree, Comput. Chem. Eng., 15 (1992), 25–33.Google Scholar
[126]Serano, C., Rodrigues, A. and Villadsen, J., Solution of partial differential equations systems by the moving finite element method, Comput. Chem. Eng., 16 (1992), 583–592.Google Scholar
[127]Shodja, H. M. and Feldkamp, J. R., Numerical analysis of sedimentation and consolidation by the moving finite element method, Int. J. Numer. Anal. Met., 17 (1993), 753–769.Google Scholar
[128]Smyth, N. F. and Hill, J. M., High order nonlinear diffusion, IMA J. Appl. Math., 40 (1988), 73–86.Google Scholar
[129]Soulaimani, A. and Saad, Y., An arbitrary Lagrangian-Eulerian finite element method for solving three-dimensional free surface flows, Comput. Method. Appl. Math., 162 (1998), 70–106.Google Scholar
[130]Staniforth, A. and Cote, J., Eliminating the interpolation associated with the semi-lagrangian scheme, Mon. Weather Rev., 114 (1986), 135–146.Google Scholar
[131]Strang, G. and Fix, G., An Analysis of the Finite Element Method, Cambridge University Press, 2nd edition, 2008.Google Scholar
[132]Stojsavljevic, J., Investigation of Waiting Times in Nonlinear Diffusion Equations Using a Moving Mesh Method, MSc dissertation, Department of Mathematics, University of Reading, UK, 2007.Google Scholar
[133]Szabo, P. and Hassager, O., Simulation of free surfaces in 3-d with the arbitrary Lagrange-Euler method, Int. J. Numer. Meth. Eng., 38 (1995), 717–734.Google Scholar
[134]Tang, T., Moving mesh computations for computational fluid dynamics, in Recent Advances in Adaptive Computations, Vol. 383 of Contemporary Mathematics, AMS, 141–173.Google Scholar
[135]Thomas, P. D. and Lombard, C. K., The geometric conservation law and its application to flow computations on moving grids, AIAA J., 17 (1979), 1030–1037.Google Scholar
[136]Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, Berlin, 2005.Google Scholar
[137]Toro, E. F. and Garcia-Navarro, P., Godunov-type methods for free-surface shallow flows: a review, J. Hydraul. Res., 45 (2007), 736–751.Google Scholar
[138]Tourigny, Y. and Baines, M. J., Analysis of an algorithm for generating locally optimal meshes for approximation by discontinuous piecewise polynomials, Math. Comput., 66 (1997), 623–650.Google Scholar
[139]Trulio, J. G. and Trigger, K. R., Numerical solution of the one-dimensional hydrodynamic equations in an arbitrary time-dependent coordinate system, Technical Report UCLR-6522, University of California Lawrence Radiation Laboratory, 1961.Google Scholar
[140]Twigger, A., Blow-up in the Nonlinear Schrodinger Equation Using an Adaptive Mesh Method, MSc dissertation, Department of Mathematics, University of Reading, UK, 2008.Google Scholar
[141]Vegt, J. J. W. van der and Ven, H. van der, Space-time discontinuous Galerkin finite element method with dynamic grid motion for inviscid compressible flows: I, general formulation, J. Comput. Phys., 182(2) (2002), 546–585.Google Scholar
[142]Vazquez, J. L., The Porous Medium Equation, Oxford Science Publications, 2007.Google Scholar
[143]Wacher, A. and Sobey, I., String gradient weighted moving finite elements in multiple dimensions with applications in two dimensions, SIAM J. Sci. Comput., 29 (2007), 459–480.Google Scholar
[144]Walkley, M. A., Gaskell, P. H., Jimack, P. K., Kelmanson, M. A. and Summers, J. L., Finite element simulation of three-dimensional free-surface flow problems, J. Sci. Comput., 24 (2005), 147–162.Google Scholar
[145]Walkley, M. A., Gaskell, P. H., Jimack, P. K., Kelmanson, M. A. and Summers, J. L., Finite element simulation of three-dimensional free-surface flow problems with dynamic contact lines, Int. J. Numer. Meth. Fluids, 47 (2005), 1353–1359.Google Scholar
[146]Wan, D. and Turek, S., Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows, J. Comput. Phys., 222 (2007), 28–56.Google Scholar
[147]Wang, L. R. and Ikeda, M., A Lagrangian description of sea ice dynamics using the finite element method, Ocean Model., 7 (2004), 21–38.Google Scholar
[148]Wells, B. V., A Moving Mesh Finite Element Method for the Numerical Solution of Partial Differential Equations and Systems, PhD thesis, Department of Mathematics, University of Reading, UK, 2005.Google Scholar
[149]Wells, B. V., Baines, M. J. and Glaister, P., Generation of Arbitrary Lagrangian-Eulerian (ALE) velocities, based on monitor functions, for the solution of compressible fluid equations, Int. J. Numer. Meth. Fluids, 47 (2005), 1375–1381.Google Scholar
[150]Wesseling, P., Computational Fluid Dynamics, Springer, 2000.Google Scholar
[151]Zhou, H. and Derby, J. J., An assessment of a parallel, finite element method for three-dimensional, moving-boundary flows driven by capillarity for simulation of viscous sintering, Int. J. Numer. Meth. Fluids, 36 (2001), 841–865.Google Scholar