The optimal accuracy of difference schemes
Authors:
Arieh Iserles and Gilbert Strang
Journal:
Trans. Amer. Math. Soc. 277 (1983), 779803
MSC:
Primary 65M10; Secondary 41A21
MathSciNet review:
694388
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Abstract: We consider difference approximations to the model hyperbolic equation which compute each new value as a combination of the known values . For such schemes we find the optimal order of accuracy: stability is possible for small if and only if . A similar bound is established for implicit methods. In this case the most accurate schemes are based on Padé approximations to near , and we find an expression for the difference ; this allows us to test the von Neumann condition . We also determine the number of zeros of in the unit circle, which decides whether the implicit part is uniformly invertible.
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 [2]
 G. Dahlquist, A special stability problem for linear multistep methods, BIT 3 (1963), 2743. MR 0170477 (30:715)
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 , Convergence and stability in the numerical integration of ordinary differential equations, Math. Scand. 4 (1956), 3353. MR 0080998 (18:338d)
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 J. W. Daniel and R. E. Moore, Computation and theory in ordinary differential equations, Freeman, San Francisco, 1970. MR 0267765 (42:2667)
 [5]
 B. L. Ehle, stable methods and Padé approximations to the exponential, Siam J. Numer. Anal. 4 (1973), 571580. MR 0331787 (48:10119)
 [6]
 B. Engquist and S. Osher, Onesided difference approximations for nonlinear conservation laws, Math. Comp. 36 (1981), 321352. MR 606500 (82c:65056)
 [7]
 A. Erdelyi, et al., Higher transcendental functions. I, McGrawHill, New York, 1953.
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 A. Iserles and M. J. D. Powell, On the acceptability of rational approximations that interpolate the exponential function, IMA J. Numer. Anal. 1 (1981), 241251. MR 641308 (83a:65015)
 [9]
 A. Iserles, Order stars and a saturation theorem for first order hyperbolics, IMA J. Numer. Anal. (to appear). MR 654266 (83d:65253)
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 , A note on Padé approximations and generalized hypergeometric functions, BIT 19 (1979), 543545. MR 559965 (81a:41028)
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 , On the generalized Padé approximations to the exponential function, Siam J. Numer. Anal. 16 (1979), 631636. MR 537277 (80i:41013)
 [12]
 P. D. Lax, On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients, Comm. Pure Appl. Math. 14 (1961), 497520. MR 0145686 (26:3215)
 [13]
 E. D. Rainville, Special functions, Macmillan, New York, 1967. MR 0107725 (21:6447)
 [14]
 G. Strang, Trigonometric polynomials and difference methods of maximum accuracy, J. Math. Phys. 41 (1962), 147154.
 [15]
 , Accurate partial difference methods. II. Nonlinear problems, Numer. Math. 6 (1964), 3746. MR 0166942 (29:4215)
 [16]
 , WienerHopf difference equations, J. Math. Mech. 13 (1964), 3746. MR 0160335 (28:3548)
 [17]
 , Implicit difference methods for initialboundary value problems, J. Math. Anal. Appl. 16 (1966), 188198. MR 0205496 (34:5323)
 [18]
 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 1939.
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 G. E. Wanner, E. Hairer and S. P. Nørsett, Order stars and stability theorems, BIT 18 (1978), 475489. MR 520756 (81b:65070)
 [20]
 O. B. Widlund, On Lax's theorem on Friedrichs type finite difference schemes, Comm. Pure Appl. Math. 24 (1971), 117123. MR 0275693 (43:1446)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198306943889
PII:
S 00029947(1983)06943889
Article copyright:
© Copyright 1983 American Mathematical Society
