David C. Arney
Abstract
We discuss mesh-moving, static mesh-regeneration, and local mesh-refinement algorithms that can be used with a finite difference or finite element scheme to solve initial-boundary value problems for vector systems of time-dependent partial differential equations in two space dimensions and time. A coarse base mesh of quadrilateral cells is moved by an algebraic mesh-movement function so as to follow and isolate spatially distinct phenomena. The local mesh-refinement method recursively divides the time step and spatial cells of the moving base mesh in regions where error indicators are high until a prescribed tolerance is satisfied. The static mesh-regeneration procedure is used to create a new base mesh when the existing one becomes too distorted. The adaptive methods have been combined with a MacCormack finite difference scheme for hyperbolic systems and an error indicator based upon estimates of the local discretization error obtained by Richardson extrapolation. Results are presented for several computational examples.
- 1 ADJERID, S., AND 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 (1986), 778-795. Google Scholar
- 2 ADJERID, S., AND FLAHERTY, J.E. A moving mesh finite element method with local refinement for parabolic partial differential equations. Comput. Methods Appl. Mech. Eng. 56 (1986), 3-26. Google Scholar
- 3 ADJERID, S., AND FLAHERTY, J.E. A local refinement finite element method for two-dimensional parabolic systems. Tech. Rep. 86-7, Dept. of Computer Science, Rensselaer Polytechnic Institute, Troy, N.Y., 1986.Google Scholar
- 4 ARNEY, D. C., BISWAS, R., AND FLAHERTY, J.E. A posteriori error estimation of adaptive finite difference schemes for hyperbolic systems. In Transactions o/the Fifth Army Conference on Applied Mathematics and Computing (West Point, N.Y., June 1987). U.S. Army Research Office, Research Triangle Park, NC, 437-458.Google Scholar
- 5 ARNEY, D. C., AND FLAHERTY, J. E. A two-dimensional mesh moving technique for time dependent partial differential equations. J. Comput. Phys. 67 (1986), 124-144. Google Scholar
- 6 ARNEY, D. C., AND FLAHERTY, J. E. An adaptive local mesh refinement method for timedependent partial differential equations. Tech. Rep. 86-10, Dept. of Computer Science, Rensselaer Polytechnic Institute, Troy, N.Y., 1986.Google Scholar
- 7 ARNE~, D. C., AND FLAltERTY, J.E. An adaptive method with mesh moving and local mesh refinement for time-dependent partial differential equations. In Transactions of the Fourth Army Conference on Applied Mathematics and Computing (Ithaca, N.Y., May 1986). U.S. Army Research Office, Research Triangle Park, NC, 1115-1141.Google Scholar
- 8 BABUSKA, I., CHANDRA, J., AND FLAHERTY, J. E., EDS. Adaptive Computational Methods for Partial Differential Equations. SIAM, Philadelphia, Pa., 1983. Google Scholar
- 9 BABUSKA, i., ZIENKIEWlCZ, O. C., GAGO, J. R., AND DE A. OLIVERA, E. R., EDS. Accuracy Estimates and Adaptive Refinements in Finite Element Computations. John Wiley, Chichester, 1986.Google Scholar
- 10 BEN-DOR, G., AND GLASS, I.I. Non-stationary oblique shock-wave reflections: Actual isopycnics and numerical experiments. AfAA J. 16 (1978), 1146-1153.Google Scholar
- 11 BEN-DOR, G., AND GLASS, I.I. Domains and boundaries of non-stationary oblique shock-wave reflections. 1. Diatomic gas. J. Fluid Mech. 92 (1979), 459-496.Google Scholar
- 12 BERGER, M. On conservation at grid interfaces. ICASE Rep. 84-43, ICASE, NASA Langley Research Center, Hampton, 1984.Google Scholar
- 13 BERGER, M., AND COLLELA, P. Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82 (1989), 64-84. Google Scholar
- 14 BERGER, M., AND OLIGER, J. Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53 (1984), 484-512.Google Scholar
- 15 BIETERMAN, M., FLAHERTY, J. E., AND MOORE, P.K. Adaptive refinement methods for nonlinear parabolic partial differential equations. In Accuracy Estimates and Adaptive Refinements in Finite Element Computations, chap. 19, I. Babuska, O. C. Zienkiewicz, J. R. Gago, and E. R. de A. Olivera, Eds., John Wiley, Chichester, 1986.Google Scholar
- 16 BRACKBILL, J. U., AND SALTZMAN, J.S. Adaptive zoning for singular problems in two dimensions. J. Comput. Phys. 46 (1982), 342-368.Google Scholar
- 17 CHAKRAVARTHY, S. R., AND OSHER, S. Computing with high-resolution upwind schemes for hyperbolic equations. In Large-Scale Computations in Fluid Mechanics, Lectures in Applied Mathematics, 22-1, B. E. Engquist, S. Osher, and R. C. J. Somerville, Eds., AMS, Providence, R.I., 1985, 57-86.Google Scholar
- 18 COYLE, J. M., FLAHERT~, J. E., AND LUDWIG, R. On the stability of mesh equidistribution strategies for time-dependent partial differential equations. J. Comput. Phys. 62 (1986), 26-39. Google Scholar
- 19 DAVIS, S.F. TVD finite difference schemes and artificial viscosity. ICASE Rep. 84-20, NASA CR 172373, ICASE, NASA Langley Research Center, Hampton, 1984.Google Scholar
- 20 DAVIS, S. F., AND FLAHERTY, J. E. An adaptive finite element method for initial-boundary value problems for partial differential equations. SIAM J. Sci. Star. Comput. 3 (1982), 6-27.Google Scholar
- 21 DORR, M.R. The approximation theory for the p-version of the finite element method, I. SIAM J, Numer. Anal. 21 (1984), 1180-1207.Google Scholar
- 22 GROPP, W.D. Local uniform mesh refinement with moving grids. SIAM J. Sci. Stat. Comput. 8 (1987), 292-304. Google Scholar
- 23 HINDMAN, R. Generalized coordinate forms of governing fluid equations and associated geometrically induced errors. AIAA J. 20 (1982), 1359-1367.Google Scholar
- 24 MACCORMACK, R.W. The effect of viscosity in hypervelocity impact cratering. AIAA Paper 69-354, 1969.Google Scholar
- 25 MCRAE, G., GOODIN, W., AND SEtNFEt.D, J. Numerical solution of the atmospheric diffusion equation for chemically reacting flows, j. Comput. Phys. 45 (1982), 1-42.Google Scholar
- 26 ODEN, J. T., STROUBOULIS, T., AND DEVLOO, P. Adaptive finite element methods for the analysis of inviscid compressible flow, I. Fast refinement/unrefinement and moving mesh methods for unstructured meshes. Comput. Methods Appl. Mech. Eng. 59 (1986), 327-362. Google Scholar
- 27 RM, M. M. Patched-grid calculations with the Euler and Navier-Stokes equations. SIAM National Meeting, Boston, Mass., July 1986.Google Scholar
- 28 RAI, M. M., AND ANDERSON, D. Grid evolution in time asymptotic problems. J. Comput. Phys. 43 (1981), 327-344.Google Scholar
- 29 THOMPSON, J. F. A survey of dynamically-adaptive grids in numerical solution of partial differential equations. Appl. Numer. Math. 1 (1985), 3-27.Google Scholar
- 30 WOODWARD, P., AND COLLELA, P. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54 (1984), 115-173.Google Scholar
Index Terms
An adaptive mesh-moving and local refinement method for time-dependent partial differential equations
Recommendations
Numerical simulations of time-dependent partial differential equations
When a time-dependent partial differential equation (PDE) is discretized in space with a spectral approximation, the result is a coupled system of ordinary differential equations (ODEs) in time. This is the notion of the method of lines (MOL), and the ...
Quasi-wavelet method for time-dependent fractional partial differential equation
In this decade, many new applications in engineering and science are governed by a series of fractional partial differential equations. In this paper, we propose a novel numerical method for a class of time-dependent fractional partial differential ...
MOVCOL4: A Moving Mesh Code for Fourth-Order Time-Dependent Partial Differential Equations
In this paper we develop and analyze a moving mesh code for the adaptive simulation of fourth-order PDEs based on collocation. The scheme is shown to enforce discrete conservation for problems written in a generalized conservation form. To demonstrate ...
Reviews
Ionel Michael Navon
The authors present a mesh-moving and local mesh refinement algorithm for time-dependent systems of partial differential equations in two dimensions that are discretized by either finite-difference or finite-element methods. The mesh-moving procedure is based on the idea of roughly following isolated nonuniformities such as shock layers or wave fronts, thus reducing dispersive errors and allowing the use of larger time steps. A nearest neighbor clustering algorithm of Berger and Oliger [1] is used in connection with this part of the algorithm. The local refinement part of the algorithm is based on an error indicator that identifies regions where a greater resolution is needed. Finer grids are created by a bisection recursive mesh refinement scheme, and error indicators are computed at each stage of the refinement. Due to severe distortion in some instances of mesh moving, a static mesh regeneration procedure is used to create a new base mesh. The authors have tested the capabilities of the adaptive procedure on three different hyperbolic systems of equations and have compared it to other adaptive mesh refinement strategies such as that of Berger and Oliger [1]. They also address issues of the computational cost, accuracy, and efficiency of the adaptive algorithm. Numerical experience indicates that the mesh moving strategy by itself and in combination with mesh refinement provides marked improvements in the solution of the three test problems. In general, however, the authors conclude that mesh moving procedures perform better alone than with refinement. The paper is well written but could profit from being more self-contained. The reader is frequently sent to technical reports or conference proceedings for details. The procedure seems to have a great potential and should prove very useful for three-dimensional problems when used with a more efficient error indicator derived from p-refinement. Its extension to parallel computing environments would also be a welcome improvement.
Access critical reviews of Computing literature here
Become a reviewer for Computing Reviews.
Comments