The Rate of Convergence of Some Difference Schemes

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Previous article Next article The Rate of Convergence of Some Difference SchemesG. W. HedstromG. W. Hedstromhttps://doi.org/10.1137/0705031PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Lars V. Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953xii+247 MR0054016 (14,857a) 0052.07002 Google Scholar[2] Mats Y. T. Apelkrans, On difference schemes for hyperbolic equations with discontinuous initial values, Math. Comp., 22 (1968), 525–539 MR0233527 (38:1848) 0164.45502 CrossrefISIGoogle Scholar[3] G. W. Hedstrom, The near-stability of the Lax-Wendroff method, Numer. Math., 7 (1965), 73–77 10.1007/BF01397974 MR0174182 (30:4389) 0131.34301 CrossrefGoogle Scholar[4] G. W. Hedstrom, Norms of powers of absolutely convergent Fourier series, Michigan Math. J., 13 (1966), 393–416 10.1307/mmj/1028999598 MR0203340 (34:3193) 0148.04401 CrossrefISIGoogle Scholar[5] M. L. Juncosa and , D. M. Young, On the order of convergence of solutions of a difference equation to a solution of the diffusion equation, J. Soc. Indust. Appl. Math., 1 (1953), 111–135 10.1137/0101007 MR0060907 (15,746d) 0053.46303 LinkGoogle Scholar[6] M. L. Juncosa and , David Young, On the Crank-Nicolson procedure for solving parabolic partial differential equations, Proc. Cambridge Philos. Soc., 53 (1957), 448–461 MR0088804 (19,583c) 0079.33804 CrossrefGoogle Scholar[7] Jaak Peetre, Espaces d'interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble), 16 (1966), 279–317 MR0221282 (36:4334) 0151.17903 CrossrefISIGoogle Scholar[8] Jaak Peetre and , Vidar Thomée, On the rate of convergence for discrete initial-value problems, Math. Scand., 21 (1967), 159–176 (1969) MR0255085 (40:8292) 0172.13902 CrossrefISIGoogle Scholar[9] S. I. Serdjukova, A study of stability in C of explicit difference schemes with constant real coefficients which are stable in $l\sb{2}$, Ž. Vyčisl. Mat. i Mat. Fiz., 3 (1963), 365–370 MR0165710 (29:2990) 0143.37901 Google Scholar[10] S. I. Serdjukova, Stability in C of linear difference schemes with constant real coefficients, Ž. Vyčisl. Mat. i Mat. Fiz., 6 (1966), 477–486 MR0205502 (34:5329) 0173.44501 Google Scholar[11] Gilbert Strang, Polynomial approximation of Bernstein type, Trans. Amer. Math. Soc., 105 (1962), 525–535 MR0141921 (25:5318) 0119.28703 CrossrefGoogle Scholar[12] Mitchell H. Taibleson, On the theory of Lipschitz spaces of distributions on Euclidean n-space. I. Principal properties, J. Math. Mech., 13 (1964), 407–479 MR0163159 (29:462) 0132.09402 ISIGoogle Scholar[13] Vidar Thomée, Stability of difference schemes in the maximum-norm, J. Differential Equations, 1 (1965), 273–292 MR0176240 (31:515) 0259.65086 CrossrefISIGoogle Scholar[14] Vidar Thomée, J. Bramble, On maximum-norm stable difference operatorsNumerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965), Academic Press, New York, 1966, 125–151 MR0210332 (35:1225) 0149.11702 Google Scholar[15] Wolfgang Wasow, On the accuracy of implicit difference aproximations to the equation of heat flow, Math. Tables Aids Comput., 12 (1958), 43–55 MR0100362 (20:6795) 0082.33301 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Multidimensional upwind hydrodynamics on unstructured meshes using graphics processing units – I. 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