zbMATH — the first resource for mathematics

An efficient dynamically adaptive mesh for potentially singular solutions. (English) Zbl 0986.65087
Summary: We develop an efficient dynamically adaptive mesh generator for time-dependent problems in two or more dimensions. The mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic partial differential equations for the mesh map. When coupled to a physical problem, the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior. The overall mesh strategy is simple to implement, avoids interpolation, and can be easily incorporated into a broad range of applications.
The efficacy of the mesh is first demonstrated by two examples of blowing-up solutions to the 2-D semilinear heat equation. These examples show that the mesh can folow with high adaptivity a finite-time singularity process. The focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2-D Boussinesq fluid, a challenging problem. The moving mesh follows effectively the flow resolves both its global features and the almost singular shear layers developed dynamically. The numerical results show the fast collapse to small scales and an exponential vorticity growth.

65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
76D17 Viscous vortex flows
PDF BibTeX Cite
Full Text: DOI
[1] Adjerid, S.; Flaherty, J.E., A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations, SIAM J. numer. anal., 23, 778, (1986) · Zbl 0612.65071
[2] Anderson, C.; Greengard, C., The vortex ring merger problem at infinite Reynolds-number, Commun. pure appl. math., 42, 1123, (1989) · Zbl 0689.76011
[3] Baines, M.J., Moving finite elements, (1994) · Zbl 0817.65082
[4] Beale, J.T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3D incompressible Euler equations, Commun. math. phys., 94, 61, (1984) · Zbl 0573.76029
[5] Bebernes, J.; Eberly, D., Mathematical problems from combustion theory, (1989) · Zbl 0692.35001
[6] Bell, J.B.; Marcus, D.L., Vorticity intensification and transition to turbulence in the 3-dimensional Euler equations, Commun. math. phys., 147, 371, (1992) · Zbl 0755.76062
[7] Berger, M.J.; Collela, P., Local adaptive mesh refinement for shock hydrodynamics, J. comput. phy., 82, 62, (1989)
[8] Brachet, M.E.; Meiron, D.I.; Orszag, S.A.; Nickel, B.G.; Morf, R.H.; Frisch, U., Small-scale structure of the taylor – green vortex, J. fluid mech., 130, 411, (1983) · Zbl 0517.76033
[9] Brachet, M.E.; Meneguzzi, M.; Vincent, A.; Politano, H.; Sulem, P.L., Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for 3-dimensional ideal flows, Phys. fluids A, 4, 2845, (1992) · Zbl 0775.76026
[10] Brackbill, J.U., An adaptive grid with directional control, J. comput. phys., 108, 38, (1993) · Zbl 0832.65132
[11] Brackbill, J.U.; Slatzman, J.S., Adaptive zoning for singular problems in two dimensions, J. comput. phys., 46, 342, (1982) · Zbl 0489.76007
[12] Budd, C.J.; Chen, S.; Russell, R.D., New self-similar solutions of the nonlinear Schrödinger equation with moving mesh computations, J. comput. phys., 152, 756, (1999) · Zbl 0942.65085
[13] Budd, C.J.; Huang, W.; Russell, R.D., Moving mesh methods for problems with blow-up, SIAM J. sci. comput., 17, 305, (1996) · Zbl 0860.35050
[14] Caflisch, R.E., Singularity formation for complex solutions of the 3D incompressible Euler equations, Physica D, 67, 1, (1993) · Zbl 0789.76013
[15] Castillo, J.E., A discrete variational grid generation method, SIAM J. sci. stat. comput., 12, 454, (1991) · Zbl 0718.65079
[16] Castillo, J.E.; Otto, J.S., A practical guide to direct optimization for planar grid-generation, Comput. math. appl., 37, 123, (1999) · Zbl 0947.65129
[17] Chang, Y.C.; Hou, T.Y.; Merriman, B.; Osher, S., A level set formulation of Eulerian interface capturing methods for incompressible fluid flows, J. comput. phys., 124, 449, (1996) · Zbl 0847.76048
[18] Chorin, A.J., The evolution of a turbulent vortex, Commun. math. phys., 83, 517, (1982) · Zbl 0494.76024
[19] Constantin, P.; Fefferman, C.; Majda, A.J., Geometric constraints on potentially singular solutions for the 3-D Euler equations, Commun. part. diff. eq., 21, 559, (1996) · Zbl 0853.35091
[20] Constantin, P.; Majda, A.J.; Tabak, E.G., Singular front formation in a model for quasigeostrophic flow, Phys. fluids, 6, 9, (1994) · Zbl 0826.76014
[21] de Boor, C., Good approximation by splines with variable knots II, (1973) · Zbl 0255.41007
[22] Dorfi, E.A.; Drury, L.O’C., Simple adaptive grids for 1-D initial value problems, J. comput. phys., 69, 175, (1987) · Zbl 0607.76041
[23] Drazin, P.G.; Reid, W.H., Hydrodynamic stability. Cambridge monographs on mechanics and applied mathematics, (1981), Cambridge Univ. Press New York · Zbl 0449.76027
[24] T. Dupont, Private communication.
[25] Dvinsky, A.S., Adaptive grid generation from harmonic maps on Riemannian manifolds, J. comput. phys., 95, 450, (1991) · Zbl 0733.65074
[26] E, W.; Shu, C.-H., Small-scale structures in Boussinesq convection, Phys. fluids, 1, 49, (1994) · Zbl 0822.76087
[27] Friedman, A; McLeod, B., Blow-up of positive solutions of semilinear heat-equations, Indiana univ. math. J., 34, 425, (1985) · Zbl 0576.35068
[28] Grauer, R.; Marliani, C.; Germaschewski, K., Adaptive mesh refinement for singular solutions of the incompressible Euler equations, Phys. rev. lett., 80, 4177, (1998)
[29] Grauer, R.; Sideris, T.C., Numerical computation of 3D incompressible ideal fluids with swirl, Phys. rev. lett., 67, 6511, (1991)
[30] Grauer, R.; Sideris, T.C., Finite time singularities in ideal fluids with swirl, Physica D, 88, 116, (1995) · Zbl 0899.76286
[31] Huang, W.; Ren, Y.; Russel, R.D., Moving mesh methods based on moving mesh partial differential equations, J. comput. phys., 113, 279, (1994) · Zbl 0807.65101
[32] Huang, W.; Ren, Y.; Russel, R.D., Moving mesh partial differential equations (MMPDEs) based on the equidistribution principle, SIAM J. numer. anal., 31, 709, (1994) · Zbl 0806.65092
[33] Huang, W.; Russell, R.D., Moving mesh strategy based on a gradient flow equation for two-dimensional problems, SIAM J. sci. comput., 20, 998, (1999) · Zbl 0956.76076
[34] Kerr, R.M., Evidence for a singularity of the 3-dimensional, incompressible Euler equations, Phys. fluids A, 5, 1725, (1993) · Zbl 0800.76083
[35] Knupp, P.; Steinberg, S., Fundamentals of grid generation, (1993), CRC Press Boca Raton
[36] R. Li, T. Tang, and, P. Zhang, Moving mesh methods in multiple dimensions based on harmonic maps, J. Comput. Phys, in press. · Zbl 0986.65090
[37] Liao, G.; Liu, F.; de la Pena, C.; Peng, D.; Osher, S., Level-set based deformation methods for adaptive grids, J. comput. phys., 159, 103, (2000) · Zbl 0958.65100
[38] Longuet-Higgins, M.S.; Cokelet, E.D., The deformation of steep surface waves on water I. A numerical method of computation, Proc. R. soc. lond. A., 350, (1976) · Zbl 0346.76006
[39] Miller, K.; Miller, R.N., Moving finite elements I, SIAM J. numer. anal., 18, 1019, (1981) · Zbl 0518.65082
[40] Petzold, L.R., Observations on an adaptive moving grid method for one-dimensional systems of partial differential equations, Appl. numer. math., 3, 347, (1987) · Zbl 0621.65123
[41] Pumir, A.; Siggia, E., Collapsing solutions to the 3-D Euler equations, Phys. fluids A, 2, 220, (1990) · Zbl 0696.76070
[42] Pumir, A.; Siggia, E.D., Development of singular solutions to the axisymmetric Euler equations, Phys. fluids A, 4, 1472, (1992) · Zbl 0825.76121
[43] Pumir, A.; Siggia, E.D., Finite-time singularities in the axisymmetric three-dimensional Euler equations, Phys. rev. lett., 68, 1511, (1992) · Zbl 0789.76016
[44] Ren, W.; Wang, X.-P., An iterative grid redistribution method for singular problems in multiple dimensions, J. comput. phys., 159, 246, (2000) · Zbl 0959.65129
[45] Shelley, M.J.; Meiron, D.I.; Orszag, S.A., Dynamic aspects of vortex reconnection of perturbed anti-parallel vortex tubes, J. fluid mech., 246, 613, (1993) · Zbl 0781.76028
[46] Thompson, J.F.; Warsi, Z.U.A.; Mastin, C.W., Numerical grid generation, (1985) · Zbl 0598.65086
[47] Winslow, A., Numerical solution of the quasi-linear Poisson equation on a nonuniform trainagle mesh, J. comput. phys., 1, 149, (1967)
[48] De Zeeuw, P.M., Matrix-dependent prolongation and restrictions in a blackbox multigrid solver, J. comput. applied math., 33, 1, (1990) · Zbl 0717.65099
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.