8th Int'l Conference on Numerical Methods in Fluid Dynamics by Krause E. (ed.)

By Krause E. (ed.)

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FIG. 13. Same as Fig. 51) is truncated after LN(x) with N = 40. Observe that the approximation is accurate for roughly 4% wavelengths of sin x. This page intentionally left blank SECTION 4 Review of Convergence Theory The fundamental problem of the numerical analysis of initial value problems is to find conditions under which UH(X, t) converges to u(x, /) as N"-* oo for some time interval 0 ^ ? ^ T and to estimate the error \\u — UN\\- The principal result is the Lax-Richtmyer equivalence theorem which states that stability is equivalent to convergence for consistent approximations to well-posed linear problems.

Laguerre expansions. (*) = 0, w(x) = e~x for 0 ^ x < o o with e~x/2n(x) bounded at x = 0 and oo. The nth eigenvalue is \n = n and the associated eigenfunction is (f>n(x) = Ln(x), the Laguerre polynomial of degree n. 30) that the Legendre expansion converges faster than algebraically as the number of terms JV-»oo. To illustrate the rate of convergence of Laguerre series, we consider the expansion of sin x: which converges for all x, 0 ^ x < oo. Since [see Erdelyi et al. (1953, p. 44*. 06polynomials per wavelength to achieve high accuracy.

Let us suppose that the Hilbert space 3€ possesses the inner product ( • , • ) • Using the inner product, we define (neglecting the complications due to boundary conditions) the adjoint L* of an operator L as that linear operator that satisfies (w, Lv)=(L*u, v) for all u, v in ffl. For the finite dimensional approximation LN, the matrix representation of L^ is the adjoint of the matrix representation of LN (see § 2). The operator LN is said to be a normal operator if LN commutes with Lfrso LNL$,= L%LN.

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