The Kuramoto–Sivashinsky equation is a prototypical chaotic nonlinear partial differential equation (PDE) in which the size of the spatial domain plays the role of a bifurcation parameter. We investigate the changing dynamics of the Kuramoto–Sivashinsky PDE by calculating the Lyapunov spectra over a large range of domain sizes. Our comprehensive computation and analysis of the Lyapunov exponents and the associated Kaplan–Yorke dimension provides new insights into the chaotic dynamics of the Kuramoto–Sivashinsky PDE, and the transition to its one-dimensional turbulence.
We prove that a general class of nonlinear, non-autonomous ODEs in Fréchet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ODE are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. In particular, our method empowers us to study approximate centre manifolds, given by solutions of ODEs that are central up to a desired, possibly nonzero precision. The main motivation is the case where the Fréchet space in question is a suitable function space, and the maps involved in an ODE in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ODE is a finite-dimensional PDE. We show that our methods apply to a relevant class of nonlinear, non-autonomous PDEs in this way.
A multiscale computational scheme is developed to use given small microscale simulations of complicated physical wave processes to empower macroscale system‐level predictions. By coupling small patches of simulations over unsimulated space, large savings in computational time are realizable. Here, we generalize the patch scheme to the case of wave systems on staggered grids in two‐dimensional (2D) space. Classic macroscale interpolation provides a generic coupling between patches that achieves consistency between the emergent macroscale simulation and the underlying microscale dynamics. Spectral analysis indicates that the resultant scheme empowers feasible computation of large macroscale simulations of wave systems even with complicated underlying physics. As an example of the scheme's application, we use it to simulate some simple scenarios of a given turbulent shallow water model.
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