Preserving Symmetries in the Proper Orthogonal Decomposition

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 inherit 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 periodic 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.