# Interdisciplinary Applied Mathematics

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(a)    (b)

FIGURE 17.14.    (a)    Comparison    of    V-velocity    at position    a —    a’    of Figure    17.12.

60 snapshots are used in both methods. (b) Comparison of V-velocity at position a — a’ of Figure 17.12. 20 snapshots and 3U+3V+3P basis functions are used in both reduced-order modeling techniques.

FIGURE 17.15. Comparison of V-velocity at position a — a’ of Figure 17.12. 3U+3V+3P basis functions are used in both reduced-order modeling techniques. 22 snapshots are used in the weighted KL method and 66 snapshots are used in the standard KL method to generate basis functions.

can produce better accuracy than a standard KL technique.

###### 17-4-2 Nonlinear Galerkin Methods

A dynamical system can be represented by a differential equation of the form

v = G(v,A),    (17.23)

where in the general case v is an element of a Hilbert space E and G(v, A) is a smooth nonlinear operator. We investigate the loss of stability of an equilibrium ve of equation (17.23) under quasi-static variation of a distinguished system parameter A. Equation (17.23) can be rewritten in the form

u = L(A)u + g(u,A),    (17.24)

where L = G„ (ve) is the linearization of the operator G at ve, the equilibrium position; g is a smooth nonlinear operator, and u = v — ve is the deviation from ve. From the point of stability we assume that equation (17.24) has an asymptotically stable equilibrium position ue = 0 for a range of parameter values A. Now A is varied quasi-statically, and it is assumed that for    A =    Ac    a    loss    of    stability    occurs    at    ue    =0. Then, two cases    exist

for which a proper dimension reduction can be performed (e ^ 1):

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