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Time-Dependent Problems 243
(10.5.14)
⎧
⎪⎨
⎪⎩
∂rn ∂rn
(−1,t) = −ζ
∂t ∂x (−1,t) − γ[rn − Lsn](−1,t) + γL[ˆqn − W](−1,t)
∂sn ∂sn
(1,t) = ζ
∂t ∂x (1,t) − γ[sn − Rrn](1,t) + γR[ˆpn − V ](1,t)
∀t ∈]0,T].
Adapting the proof of stability to this new system, we can bound rn(·,t)w, sn(·,t)w,
t ∈]0,T], by the initial data and by the errors |ˆpn − V |(1,t), |ˆqn − W |(−1,t), t ∈]0,T].
The final error estimate follows from the triangle inequalities
pn − V w ≤ rnw + ˆpn − V w, qn − W w ≤ snw + ˆqn − W w.
We finally observe that the numerical experiments are in general more accurate
using (10.5.9) in place of (10.5.7), though the expression of the constant γ is known
only in the Legendre case.
Other polynomial approximations are examined in gottlieb, gurzburger and
turkel(1982) and tal-ezer(1986b). Techniques based on spectral methods are under
development for the numerical discretization of nonlinear hyperbolic systems.
10.6 Time discretization
In the previous sections we analyzed polynomial approximations, with respect to the
space variable x, of various time-dependent partial differential equations. In order
to implement our algorithms we also need to discretize the time variable t. Although
polynomials can be used to get a global approximation in the time interval [0,T], T > 0
(see morchoisne(1979), tal-ezer(1986a) and tal-ezer(1989)), finite-differences are
generally preferred. Much has been written on this subject. We refer the reader to