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Approximation Of Stationary Statistical Properties Of Dissipative Dynamical Systems: Time Discretization, Xiaoming Wang
Approximation Of Stationary Statistical Properties Of Dissipative Dynamical Systems: Time Discretization, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
We consider temporal approximation of stationary statistical properties of dissipative infinite-dimensional dynamical systems. We demonstrate that stationary statistical properties of the time discrete approximations, i.e., numerical scheme, converge to those of the underlying continuous dissipative infinite-dimensional dynamical system under three very natural assumptions as the time step approaches zero. the three conditions that are sufficient for the convergence of the stationary statistical properties are: (1) uniform dissipativity of the scheme in the sense that the union of the global attractors for the numerical approximations is pre-compact in the phase space; (2) convergence of the solutions of the numerical scheme to …
Approximating Stationary Statistical Properties, Xiaoming Wang
Approximating Stationary Statistical Properties, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
It is well-known that physical laws for large chaotic dynamical systems are revealed statistically. Many times these statistical properties of the system must be approximated numerically. the main contribution of this manuscript is to provide simple and natural criterions on numerical methods (temporal and spatial discretization) that are able to capture the stationary statistical properties of the underlying dissipative chaotic dynamical systems asymptotically. the result on temporal approximation is a recent finding of the author, and the result on spatial approximation is a new one. Applications to the infinite Prandtl number model for convection and the barotropic quasi-geostrophic model are …
A Semi-Implicit Scheme For Stationary Statistical Properties Of The Infinite Prandtl Number Model, Wenfang Cheng, Xiaoming Wang
A Semi-Implicit Scheme For Stationary Statistical Properties Of The Infinite Prandtl Number Model, Wenfang Cheng, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
We propose a semisecret in time semi-implicit numerical scheme for the infinite Prandtl model for convection. Besides the usual finite time convergence, this scheme enjoys the additional highly desirable feature that the stationary statistical properties of the scheme converge to those of the infinite Prandtl number model at vanishing time stop. One of the key characteristics of the scheme is that it preserves the dissipativity of the infinite Prandtl number model uniformly in terms of the time stop. So far as wo know, this is the first rigorous result on convergence of stationary statistical properties of numerical schemes for infinite …