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An improvement of a recent Eulerian method for solving PDEs on general geometries. (English) Zbl 1122.65073

The author modifies the method introduced by M. Bertalmio, L.-T. Cheng, S. Osher, and G. Sapiro [J. Comput. Phys. 174, No. 2, 759–780 (2001; Zbl 0991.65055)] for solving evolution partial differential equations (PDEs) on codimension-one surfaces in \(\mathbb{R}^N\) by changing the Eulerian representation to include effects due to surface curvature, and presents numerical examples that include convergence tests in neighborhoods of the surface that shrink with the grid size. This modified PDE has the very useful property that any solution which is initially constant perpendicular to the surface remains so at later times.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35K65 Degenerate parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Citations:

Zbl 0991.65055

Software:

ITPACK; ITPACK 2C
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Full Text: DOI

References:

[1] Adalsteinsson D., and Sethian J.A. (1995). A fast level set method for propagating interfaces. J. Comput. Phys. 118(2): 269–277 · Zbl 0823.65137 · doi:10.1006/jcph.1995.1098
[2] Adalsteinsson D., and Sethian J.A. (2003). Transport and diffusion of material quantities on propagating interfaces via level set methods. J. Comput. Phys. 185(1): 271–288 · Zbl 1047.76093 · doi:10.1016/S0021-9991(02)00057-8
[3] Allen S.M., and Cahn J.W. (1979). A macroscopic theory for antiphase boundary motion and its application to antiphase domain coarsing. Acta. Metal. 27:1085–1095 · doi:10.1016/0001-6160(79)90196-2
[4] Bertalmío M., Cheng L.T., Osher S., and Sapiro G. (2001). Variational problems and partial differential equations on implicit surfaces. J. Comput. Phys. 174(2):759–780 · Zbl 0991.65055 · doi:10.1006/jcph.2001.6937
[5] Burger, M. (2005). Finite element approximation of elliptic partial differential equations on implicit surfaces. UCLA CAM Report 05-46, August 2005
[6] Cahn J.W., Mallet-Paret J., and Van Vleck E.S. (1998). Traveling wave solutions for systems of ODEs on a two-dimensional spatial lattice. SIAM J. Appl. Math. 59(2): 455–493 · Zbl 0917.34052 · doi:10.1137/S0036139996312703
[7] Carpio A., and Bonilla L.L. (2003). Depinning transitions in discrete reaction-diffusion equations. SIAM J. Appl. Math. 63(3): 1056–1082 · Zbl 1035.34058 · doi:10.1137/S003613990239006X
[8] Caselles, V., Igual, L., and Sander, O. An axiomatic approach to scalar data interpolation on surfaces preprint. · Zbl 1092.65007
[9] Chopp D.L. (2001). Some improvements of the fast marching method. SIAM J. Sci. Comput. 23(1): 230–244 · Zbl 0991.65105 · doi:10.1137/S106482750037617X
[10] Clarenz U., Diewald U., and Rumpf M. (2004). Processing textured surfaces via anisotropic geometric diffusion. IEEE Trans. Image Process. 13(2): 248–261 · Zbl 05452838 · doi:10.1109/TIP.2003.819863
[11] Clarenz, U., Rumpf, M., and Telea, A. (2004). Finite elements on point based surfaces. Comput. Graphics, to appear.
[12] Diewald U., Preußer T., and Rumpf M. (2000). Anisotropic diffusion in vector field visualization on Euclidean domains and surfaces. IEEE Trans. Visualization Comput. Graphics 6:139–149 · Zbl 05108171 · doi:10.1109/2945.856995
[13] do Carmo M.P. (1976). Differential Geometry of Curves and Surfaces. Prentice-Hall Inc., Englewood Cliffs, N.J. Translated from the Portuguese · Zbl 0326.53001
[14] Dorsey J. and Hanrahan P. (2000). Digital materials and virtual weathering. Sci. Am. 282(2): 46 · doi:10.1038/scientificamerican0200-64
[15] Evans L.C. (1998). Partial differential equations. In Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, RI · Zbl 0902.35002
[16] Evans L.C., Soner H.M., and Souganidis P.E. (1992). Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45(9): 1097–1123 · Zbl 0801.35045 · doi:10.1002/cpa.3160450903
[17] Evans L.C., and Spruck J. (1992). Motion of level sets by mean curvature II. Trans. Amer. Math. Soc. 330(1): 321–332 · Zbl 0776.53005 · doi:10.2307/2154167
[18] Frisken, S. F., Perry, R. N, Rockwood, A. and Jones, T. (2000). Adaptively sampled fields: A general representation of shape for computer graphics. ACM SIGGRAPH
[19] Glasner K. (2003). A diffuse interface approach to Hele-Shaw flow. Nonlinearity 16(1): 49–66 · Zbl 1138.76340 · doi:10.1088/0951-7715/16/1/304
[20] Greer, J. B., Bertozzi, A. L., and Sapiro, G. (2005). Fourth order partial differential equations on general geometries. UCLA CAM Report 05-17, March 2005
[21] Halpern D., Jensen O.E., and Grotberg J.B. (1998). A theoretical study of surfactant and liquid delivery into the lung. J. Appl. Physiol. 85:333–352
[22] Hofer, M., and Pottmann, H. (2004): Energy-minimizing splines in manifolds. ACM Trans. Graphics.
[23] Hoppe, H., and Eck, M. (1996). Automatic reconstruction of b-spline surfaces of arbitrary topological type. ACM SIGGRAPH
[24] Kincaid, D., Respess, J., and Young, D. Itpack 2c: A fortran package for solving large sparse linear systems by adaptive accelerated iterative methods. http://rene.ma.utexas.edu/CNA/ITPACK/. · Zbl 0485.65025
[25] Krishnamurthy, V., and Levoy, M. (1996). Fitting smooth surfaces to dense polygon meshes. ACM SIGGRAPH, 313–324
[26] Lax P., and Wendroff B. (1960). Systems of conservation laws. Commun. Pure Appl. Math. 13:217–237 · Zbl 0152.44802 · doi:10.1002/cpa.3160130205
[27] Mémoli F., and Sapiro G. (2001). Fast computation of weighted distance functions and geodesics on implicit hyper-surfaces. J. Comput. Phys. 173(2): 730–764 · Zbl 0991.65018 · doi:10.1006/jcph.2001.6910
[28] Mémoli F., Sapiro G., and Thompson P. (2004). Implicit brain imaging. Human Brain Mapping 23:179–188
[29] Myers T.G. and Charpin J.P.F. (2004). A mathematical model for atmospheric ice accretion and water flow on a cold surface. Int. J. Heat Mass Trans. 47(25): 5483–5500 · Zbl 1078.76074 · doi:10.1016/j.ijheatmasstransfer.2004.06.037
[30] Myers T.G., Charpin J.P.F., and Chapman S.J. (2002). The flow and solidification of a thin fluid film on an arbitrary three-dimensional surface. Phys. Fluids 14(8): 2788–2803 · Zbl 1185.76270 · doi:10.1063/1.1488599
[31] Osher, S., and Fedkiw, R. (2003). Level set methods and dynamic implicit surfaces. In Applied Mathematical Sciences, Vol. 153 Springer-Verlag, New York · Zbl 1026.76001
[32] Peng D., Merriman B., Osher S., Zhao H.-K., and Kang M. (1999). A PDE-based fast local level set method. J. Comput. Phys. 155(2): 410–438 · Zbl 0964.76069 · doi:10.1006/jcph.1999.6345
[33] Richtmyer R.D., and Morton K.W. (1994). Difference Methods for Initial-Value Problems 2nd ed. Robert E. Krieger Publishing Co. Inc., Malabar, FL · Zbl 0824.65084
[34] Sethian, J. A. (1999). Level set methods and fast marching method. Evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials sciences. In Cambridge Monographs on Applied and Computational Mathematics. Vol. 3, 2nd ed., Cambridge University Press, Cambridge · Zbl 0973.76003
[35] Simon, L. (1983). Lectures on geometric measure theory. In Proceedings of the Centre for Mathematical Analysis, Australian National University, Vol. 3. Australian National University Centre for Mathematical Analysis, Canberra. · Zbl 0546.49019
[36] Strang G. (1968). On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5:506–517 · Zbl 0184.38503 · doi:10.1137/0705041
[37] Taylor, M. E. Partial differential equations. I. In Applied Mathematical Sciences, Vol. 115. Springer-Verlag, New York, Basic theory · Zbl 0869.35001
[38] Toga A. (1998). Brain Warping. Academic Press, New York
[39] Turk, G. (1991). Generating textures on arbitrary surfaces using reaction-diffusion. Comput. Graphics, 25(4).
[40] Vollmayr-Lee, B. P., and Rutenberg, A. D. (2003). Fast and accurate coarsening simulation with an unconditionally stable time step. Phys. Rev. E, 68.
[41] Witkin A., and Kass M. (1991). Reaction-diffusion textures. Computer Graphics (SIGGRAPH) 25(4): 299 · doi:10.1145/127719.122750
[42] Xu, J., Li, Z., Lowengrub, J., and Zhao, H.-K. A level set method for interfacial flows with surfactant. preprint. · Zbl 1161.76548
[43] Xu J., and Zhao H.-K. (2003). An Eulerian formulation for solving partial differential equations along a moving interface. J. Sci. Comput. 19(1–3): 573–594 Special issue in honor of the sixtieth birthday of Stanley Osher · Zbl 1081.76579 · doi:10.1023/A:1025336916176
[44] Yngve, G., and Turk, G. (1999). Creating smooth implicit surfaces from polygonal meshes. Technical Report GIT-GVU-99-42, Graphics, Visualization, and Usability Center. Georgia Institute of Technology
[45] Zhao H.-K., Chan T., Merriman B. and Osher S. (1996). A variational level set approach to multiphase motion. J. Comput. Phys. 127(1): 179–195 · Zbl 0860.65050 · doi:10.1006/jcph.1996.0167
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