Numerical Algorithms for Stationary Statistical Properties of Dissipative Dynamical Systems
It is well-known that physical laws for large chaotic dynamical systems are revealed statistically. the main concern of this manuscript is numerical methods for dissipative chaotic infinite dimensional dynamical systems that are able to capture the stationary statistical properties of the underlying dynamical systems. We first survey results on temporal and spatial approximations that enjoy the desired properties. We then present a new result on fully discretized approximations of infinite dimensional dissipative chaotic dynamical systems that are able to capture asymptotically the stationary statistical properties. the main ingredients in ensuring the convergence of the long time statistical properties of the numerical schemes are: (1) uniform dissipativity of the scheme in the sense that the union of the global attractors of the numerical approximations is pre-compact in the phase space; (2) convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval [0, 1] modulo an initial layer, uniformly with respect to initial data from the union of the global attractors. the two conditions are reminiscent of the Lax equivalence theorem where stability and consistency are needed for the convergence of a numerical scheme. Applications to the complex Ginzburg-Landau equation and the two-dimensional Navier-Stokes equations in a periodic box are discussed.
X. Wang, "Numerical Algorithms for Stationary Statistical Properties of Dissipative Dynamical Systems," Discrete and Continuous Dynamical Systems- Series A, vol. 36, no. 8, pp. 4599 - 4618, American Institute of Mathematical Sciences (AIMS), Aug 2016.
The definitive version is available at https://doi.org/10.3934/dcds.2016.36.4599
Mathematics and Statistics
Keywords and Phrases
Complex Ginzburg-Landau Equation; Dissipative System; Finite Difference; Global Attractor; Invariant Measure; Lax Equivalence Type Conditions; Navier-Stokes Equations; Spatial Discretization; Spectral Collocation Method; Stationary Statistical Property; Time Discretization; Uniformly Dissipative Schemes
International Standard Serial Number (ISSN)
Article - Journal
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01 Aug 2016