A Test of Moving Mesh Refinement for 2-D Scalar Hyperbolic Problems (1980)

Previous article Next article A Test of Moving Mesh Refinement for 2-D Scalar Hyperbolic ProblemsWilliam D. GroppWilliam D. Gropphttps://doi.org/10.1137/0901012PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe present an algorithm for numerically solving hyperbolic equations with moving shock fronts. The central idea is a moving, finer mesh which follows the shock. Computational results are presented.[1] Gerald Browning, , Heinz-Otto Kreiss and , Joseph Oliger, Mesh refinement, Math. Comp., 27 (1973), 29–39 48:12861 0256.65041 CrossrefISIGoogle Scholar[2] Melvyn Ciment and , Roland A. Sweet, Mesh refinements for parabolic equations, J. Computational Phys., 12 (1973), 513–525 10.1016/0021-9991(73)90102-2 53:12027 0264.65054 CrossrefISIGoogle Scholar[3] John Guckenheimer, Shocks and rarefactions in two space dimensions, Arch. Rational Mech. Anal., 59 (1975), 281–291 10.1007/BF00251604 52:8668 0329.35043 CrossrefISIGoogle Scholar[4] A. Harten and , G. Zwas, Self-adjusting hybrid schemes for shock computations, J. Computational Phys., 9 (1972), 568–583 10.1016/0021-9991(72)90012-5 46:8449 0244.76033 CrossrefISIGoogle Scholar[5] D. C. L. Lam and , R. B. Simpson, R. Vichnevetsky, Modeling coastal effluent transport using a variable finite difference grid, Advances in Computer Methods for Partial Differential Equations II, IMACS(AICA), 1977, 294–300 Google Scholar[6] R. MacCormack, Numerical Solutions of the Interaction of a Shock Wave with a Laminar Boundary Layer, Lecture Notes on Physics, Vol. 8, Springer-Verlag, New York, 1971, M. Holt, ed. 0228.76098 CrossrefGoogle Scholar[7] Joseph Oliger, Hybrid difference methods for the initial boundary-value problem for hyperbolic equations, Math. Comp., 30 (1976), 724–738 55:1747 0345.65049 CrossrefISIGoogle Scholar[8] G. B. Whitham, Linear and nonlinear waves, Wiley-Interscience [John Wiley & Sons], New York, 1974xvi+636 58:3905 0373.76001 Google Scholar[9] N. N. Yanenko, , V. D. Lisseikin and , V. M. Kovenia, The method of the solution of gas dynamical problems in moving meshesComputing methods in applied sciences and engineering (Proc. Third Internat. Sympos., Versailles, 1977), II, Lecture Notes in Phys., Vol. 91, Springer, Berlin, 1979, 48–61 80g:76037 0398.76059 CrossrefGoogle ScholarKeywordshyperbolic partial differential equationsfinite differencesmesh refinementshocks Previous article Next article FiguresRelatedReferencesCited ByDetails Incomplete mixing in porous media: Todd-Longstaff upscaling approach versus a dynamic local grid refinement methodComputational Geosciences, Vol. 23, No. 2 | 11 December 2018 Cross Ref Numerical Solution of Multidimensional Hyperbolic PDEs Using Defect Correction on Adaptive GridsJournal of Scientific Computing, Vol. 69, No. 2 | 20 April 2016 Cross Ref Shock capturing by anisotropic diffusion oscillation reductionComputer Physics Communications, Vol. 144, No. 3 | 1 Apr 2002 Cross Ref Computational Methods for Continuum Models of Platelet AggregationJournal of Computational Physics, Vol. 151, No. 2 | 1 May 1999 Cross Ref Analysis of the Implicit Euler Local Uniform Grid Refinement MethodR. A. 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