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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