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Adaptive mesh refinement for hyperbolic partial differential equations. (English) Zbl 0536.65071
An adaptive numerical method for hyperbolic partial differential equations is presented. Based upon Richardson-type estimates of the local truncation error, refined grids are created or removed during the process. The composite grid consists of a hierarchy of rotated uniform rectangular refined meshes. This makes it possible to approximate discontinuities such as moving shock fronts with arbitrary orientations. The finer grids have also smaller step sizes in time thus maintaining the ratio of space/time differencing. Various integrating schemes can be used on component grids. The interaction between grids is performed by injections and interpolations. Data structures for the composite grid are briefly but clearly described. Numerical examples in one and two space dimensions show that the adaptive method is superior to the use of a conventional uniform grid both in the terms of computational time and approximation of the solution.
Reviewer: J. Mandel

MSC:
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35L45 Initial value problems for first-order hyperbolic systems
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References:
[1] Babuska, I.; Rheinboldt, W., SIAM J. numer. anal., 15, 736, (1978)
[2] Bank, R., A multi-level iterative method for nonlinear elliptic equations, (), 1
[3] Berger, M., ()
[4] Bolstad, J., ()
[5] Brandt, A., Math. comput., 31, 333, (1977)
[6] Ciment, M., Math. comput., 25, 219, (1971)
[7] Cramer, H., Mathematical methods of statistics, (1951), Princeton Univ. Press Princeton, N. J
[8] Davis, S.; Flaherty, J., SIAM J. sci. statist. comput., 3, 6, (1982)
[9] Duda, R.; Hart, P., Pattern classification and scene analysis, (1974), Wiley New York
[10] Dwyer, H.; Kee, R.; Sanders, B., Aiaa j., 18, 1205, (1980)
[11] Gannon, D., Self adaptive methods for parabolic partial differential equations, ()
[12] Gennery, D., Object detection and measurement using stereo vision, (), 320
[13] Gotlieb, D.; 0rszag, S., Numerical analysis of spectral methods: theory and applications, (1977), Soc. Ind. Appl. Math Philadelphia
[14] Gropp, W.D., SIAM J. sci. statist. comput., 1, 191, (1980)
[15] Gustafsson, B., Math. comput., 29, 396, (1975)
[16] Harten, A.; Hyman, J., J. comput. phys., 50, 2, 235, (1983)
[17] Hedstrom, G.W., Math. comput., 29, 964, (1975)
[18] Hartigan, J., Clustering algorithms, (1973), Academic Press New York
[19] Jameson, A., Comm. pure appl. math., 27, 283, (1974)
[20] Knuth, D., ()
[21] Kreiss, B., SIAM J. sci. statist. comput., 4, 270, (1983)
[22] Miller, K.; Miller, R., SIAM J. numer. anal., 18, 1019, (1981)
[23] {\scJ. Oliger}, to appear.
[24] Oliger, J., Approximate methods for atmospheric and oceanographic circulation problems, (), 171
[25] Pereyra, V.; Sewell, E., Numer. math., 23, 261, (1975)
[26] Sherman, A.; Saeger, M., An approach to automatic software for parabolic partial differential equations, (), 88
[27] Simpson, R.B., Automatic local refinement for irregular rectangular meshes, () · Zbl 0419.65010
[28] Sod, G., J. comput. phys., 27, 1, (1978)
[29] Starius, G., Numer. math., 35, 241, (1980)
[30] Viviand, H., Rech. aerospat., 1, 65, (1974)
[31] Winkler, K.H., A numerical procedure for the calculation of nonsteady spherical shock fronts with radiation, ()
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