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246 Polynomial Approximation of Differential Equations E4,3200 with E6,3200). The other is the error in the time variable, which only behaves like h := 1/m. Indeed, except for the case n = 4, En,m is halved when m is doubled. The reader will note that En,m does not decrease for n ≥ 6. This is because the time-step h is not small enough to compete with the good accuracy in space, thus, the global error is dominated by the time-discretization error. The situation is different for the case n = 4, where the space resolution is poor and the results cannot improve by reducing the time-step. In our example, relation (10.6.4) is not satisfied when n = 10 and m = 200. This is the reason why the error E10,200 is rather high. m E4,m E6,m E8,m E10,m 200 .5782 × 10 −3 .2426 × 10 −3 .2427 × 10 −3 .3458 × 10 28 400 .5546 × 10 −3 .1212 × 10 −3 .1213 × 10 −3 .1213 × 10 −3 800 .5522 × 10 −3 .6069 × 10 −4 .6066 × 10 −4 .6066 × 10 −4 1600 .5534 × 10 −3 .3055 × 10 −4 .3033 × 10 −4 .3032 × 10 −4 3200 .5545 × 10 −3 .1576 × 10 −4 .1517 × 10 −4 .1516 × 10 −4 Table 10.6.1 - Errors relative to the Chebyshev collocation approximation of problem (10.2.5), (10.2.6), (10.2.7). The above experiment shows that, in order to retain the good approximation prop- erties of spectral methods, the time-step should be very small. This results in a large number of steps. Although the FFT (see section 4.3) speeds up matrix-vector multipli- cations in the Chebyshev case, the global computational cost can still be very high. An alternative is to use more accurate time-discretization methods, such as Runge-Kutta, Adams-Bashforth or Du Fort-Frankel methods (see jain(1984)). By these techniques, the error decays as a fixed integer power of the time-step. For instance, experiments based on equation (10.3.20), using a fourth-order Runge-Kutta method, are discussed in solomonoff and turkel(1989). Like the Euler method, the schemes mentioned above
Time-Dependent Problems 247 are explicit. This implies that they can be used only if h satisfies an inequality of the type (10.6.4) (absolute stability condition). Indications for the choice of the parameter h are given in gottlieb and tadmor(1990) for spectral approximations of first-order partial differential equations. Unfortunately, in most of the practical applications, these stability conditions are considered to be very restrictive, especially when D is related to the space-discretization of second-order differential operators. In this case, according to (8.3.6), the maximum time-step ˜ h in (10.6.4) is required to be proportional to 1/n 4 . Indeed, this limitation is quite severe, even when n is not too large. For first-order problems, kosloff and tal-ezer(1989) propose a Chebyshev-like spectral method with a weaker restriction on ˜ h. No restrictions are in general required for implicit methods, such as the backward Euler method. This is obtained by modifying (10.6.2) to be (10.6.5) ⎧ ⎨ ⎩ ¯p (j) h := ¯p (j−1) h ¯p (0) h := ¯p0. + hD¯p (j) h + h ¯ f(tj) 1 ≤ j ≤ m, The scheme (10.6.5) is unconditionally stable (i.e. no limitations on h are required), provided the eigenvalues of D have a negative real part. In addition, we still have (10.6.3), i.e., the method is first-order accurate. At each step, the new vector ¯p (j) h , 1 ≤ j ≤ m, is computed by solving a n ×n linear system whose matrix is I −hD. We refer to section 7.6 for the numerical treatment of this system. The algorithm is now more costly, but we are allowed to choose a larger time-step. This results in a loss of accuracy, which is moderate, however, for solutions that have a slow time variation. Further theoretical results and experiments are discussed in mercier(1982) and mercier(1989) for first-order problems, and canuto and quarteroni(1987) for hy- perbolic systems. Interesting comments are given in trefethen and trummer(1987) and trefethen(1988), where numerical tests show the sensitivity to rounding errors of certain explicit methods, when coupled with spectral approximations. Similar techniques apply to nonlinear equations. For example, the Burgers equation (see section 10.4) can be discretized by introducing a sequence of polynomials p (j) n ∈ P 0 n,
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PREFACE This book is devoted to the
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CONTENTS 1. Special families of pol
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Contents ix 7. Derivative Matrices
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1 SPECIAL FAMILIES OF POLYNOMIALS A
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Special Families of Polynomials 3 n
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Special Families of Polynomials 5 B
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Special Families of Polynomials 7 1
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Special Families of Polynomials 9 (
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Special Families of Polynomials 11
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2 ORTHOGONALITY Inner products and
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Orthogonality 23 The function w is
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Orthogonality 25 As we promised, we
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Orthogonality 27 Let us now define
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Orthogonality 29 Therefore, one obt
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Orthogonality 31 From (2.3.8), we g
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Orthogonality 33 only mention the r
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3 NUMERICAL INTEGRATION As we shall
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Numerical Integration 37 In the cas
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Numerical Integration 39 (3.1.9) z
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Numerical Integration 41 We give th
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Numerical Integration 43 where 1
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Numerical Integration 45 (3.2.10)
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Numerical Integration 47 This means
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Numerical Integration 49 (3.4.2) w
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Numerical Integration 51 3.5 Gauss-
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Numerical Integration 53 The proof
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Numerical Integration 55 3.7 Clensh
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Numerical Integration 57 the first
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Numerical Integration 59 (3.8.14) p
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Numerical Integration 61 (3.9.5) p
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Numerical Integration 63 (3.10.5)
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66 Polynomial Approximation of Diff
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100 Polynomial Approximation of Dif
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Page 305 and 306: REFERENCES adams r.(1975), Sobolev
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References 297 friedman a.(1959), F
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References 299 jackson d.(1930), Th
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References 301 pavoni d.(1988), Sin
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References 303 vainikko g.m.(1964),
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