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Summary

and with xn = b for n [b 2h, b], we have [via (9.105b)] 1 1 [3f (xn ) 4f (xn 1 ) + f (xn 2 )] + h2 f (3) ( n ). 2h 3 (9.106b) Thus, (9.104) becomes (approximate corrected trapezoidal rule) f (1) (xn ) = f (1) (b) = TC (n) = T (n) h [3f (xn ) 4f (xn 1 ) + f (xn 2 ) + 3f (x0 ) 4f (x1 ) + f (x2 )] 24 (9.107)

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A promising route for developing accurate numerical methods which can also be applicable to engineering flows, is the non-linear monotone schemes in the context of finite volume methods The state of the art of these schemes aiming at satisfying the aforementioned properties are also referred to as high-resolution and high-order methods These methods emerge from the theory of hyperbolic conservation laws and the original Godunov method (4) The latter is the forerunner of all the well known approximate Riemann solvers, eg (7), (8), upwind Godunov schemes, eg (6), (10), (11), (14), TVD methods, eg (9), (12), (13), (15), and of schemes with order of accuracy higher than two (see discussion in Section 124) 12.

But x = 0 as it is an eigenvector of A, and also a = 0 Immediately, = , contradicting our assumption that these eigenvalues are distinct Thus, y = ax is impossible; that is, we have proved that y is independent of x Theorem 112 leads us to the next theorem Theorem 113: If A Cn n has n distinct eigenvalues, then A has n linearly independent eigenvectors Proof Uses mathematical induction (eg, Stewart [6]) We have already seen the following theorem Theorem 114: If A Rn n and A = AT , then all eigenvalues of A are realvalued Proof See Hill [1], or see the appropriate footnote in 4 In addition to this theorem, we also have the following one Theorem 11.

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5: If A Rn n , and if A = AT , then eigenvectors corresponding to distinct eigenvalues of A are orthogonal Proof Suppose that ( , x) and ( , y) are eigenpairs of A with = We wish to show that x T y = y T x = 0 (recall De nition 16) Now x = Ax = AT x so that y T x = y T AT x = (Ay)T x = y T x, implying that ( )y T x = 0 But = so that y T x = 0, that is, x y Theorem 115 states that eigenspaces corresponding to distinct eigenvalues of a symmetric, real-valued matrix form mutually orthogonal vector subspaces of Rn Any vector from one eigenspace must therefore be orthogonal to any eigenvector from another eigenspace If we recall De nition 11.

3 High resolution High-resolution schemes are those with the following properties (9), (17): (i) Provide at least second order of accuracy in smooth areas of the flow; (ii) Produce numerical solutions free from spurious oscillations; and (iii) In the case of discontinuities, the number of grid points in the transition zone containing the shock wave is smaller in comparison with that of first-order monotone methods High-resolution can be realized by numerically reconstructing the variables at the cell faces of a computational volume via the solution of a Riemann problem This can be achieved by using Godunov-type methods which offer an ingenious computational framework for developing schemes that adjust the amount of numerical dissipation locally, ie at the cell faces, in order to maintain monotonicity and conservation.

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