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Preserving symmetries in the proper orthogonal decomposition. (English) Zbl 0774.65084

Authors’ summary: The proper orthogonal decomposition (POD) (also called Karhunen-Loève expansion) has been recently used in turbulence to derive optimally fast converging bases of spatial functions, leading to efficient finite truncations. Whether a finite number of these modes can be used in numerical simulations to derive an “accurate” finite set of ordinary differential equations, over a certain range of bifurcation parameter values, still remains an open question.
It is shown here that a necessary condition for achieving this goal is that the truncated system inherits the symmetry properties of the original infinite-dimensional system. In most cases, this leads to a systematic involvement of the symmetry group in deriving a new expansion basis called the symmetric POD basis. The Kuramoto-Sivashinsky equation with period boundary conditions is used as a paradigm to illustrate this point of view. However, the conclusion is general and can be applied to other equations, such as the Navier-Stokes equations, the complex Ginzburg-Landau equation, and others.

MSC:

65Z05 Applications to the sciences
76F20 Dynamical systems approach to turbulence
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C40 Ordinary differential equations and systems on manifolds
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