Long-time dynamics of 2d double-diffusive convection: analysis and/of numerics

We consider a two-dimensional model of double-diffusive convection and its time discretisation using a second-order scheme (based on backward differentiation formula for the time derivative) which treats the non-linear term explicitly. Uniform bounds on the solutions of both the continuous and discrete models are derived (under a timestep restriction for the discrete model), proving the existence of attractors and invariant measures supported on them. As a consequence, the convergence of the attractors and long time statistical properties of the discrete model to those of the continuous one in the limit of vanishing timestep can be obtained following established methods.

[1]  Jack K. Hale,et al.  Infinite dimensional dynamical systems , 1983 .

[2]  Xiaoming Wang,et al.  An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier–Stokes equations , 2011, Numerische Mathematik.

[3]  Florentina Tone,et al.  Multivalued attractors and their approximation: applications to the Navier–Stokes equations , 2011, Numerische Mathematik.

[4]  Endre Süli,et al.  Upper semicontinuity of attractors for linear multistep methods approximating sectorial evolution equations , 1995 .

[5]  Namkwon Kim,et al.  Large Friction Limit and the Inviscid Limit of 2D Navier-Stokes Equations Under Navier Friction Condition , 2009, SIAM J. Math. Anal..

[6]  Herbert E. Huppert,et al.  Double-diffusive convection , 1981, Journal of Fluid Mechanics.

[7]  P. Garaud,et al.  A NEW MODEL FOR MIXING BY DOUBLE-DIFFUSIVE CONVECTION (SEMI-CONVECTION). I. THE CONDITIONS FOR LAYER FORMATION , 2011, 1112.4819.

[8]  R. A. Wentzell,et al.  Hydrodynamic and Hydromagnetic Stability. By S. CHANDRASEKHAR. Clarendon Press: Oxford University Press, 1961. 652 pp. £5. 5s. , 1962, Journal of Fluid Mechanics.

[9]  M. Holland,et al.  Thermohaline Circulation: High-Latitude Phenomena and the Difference Between the Pacific and Atlantic , 1999 .

[10]  Raymond W. Schmitt,et al.  DOUBLE DIFFUSION IN OCEANOGRAPHY , 1994 .

[11]  Bifurcation and stability of two-dimensional double-diffusive convection , 2006, physics/0611111.

[12]  G. Burton.  Sobolev Spaces , 2013 .

[13]  Roger Temam,et al.  Attractors for the Be´nard problem: existence and physical bounds on their fractal dimension , 1987 .

[14]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[15]  C. F. Chen,et al.  Double-Diffusive Convection: A report on an Engineering Foundation Conference , 1984, Journal of Fluid Mechanics.

[16]  Bounds on double-diffusive convection , 2005, Journal of Fluid Mechanics.

[17]  Xiaoming Wang,et al.  Approximation of stationary statistical properties of dissipative dynamical systems: Time discretization , 2010, Math. Comput..

[18]  Carl Wunsch,et al.  What Is the Thermohaline Circulation? , 2002, Science.

[19]  Roger Temam,et al.  Navier–Stokes Equations and Nonlinear Functional Analysis: Second Edition , 1995 .

[20]  N. Ju.  On the global stability of a temporal discretization scheme for the Navier-Stokes equations , 2002 .

[21]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[22]  R. Temam.  Navier-Stokes Equations and Nonlinear Functional Analysis , 1987 .

[23]  Endre Süli,et al.  Approximation of the global attractor for the incompressible Navier–Stokes equations , 2000 .

[24]  Xiaoming Wang,et al.  Long Time Stability of a Classical Efficient Scheme for Two-dimensional Navier-Stokes Equations , 2011, SIAM J. Numer. Anal..

[25]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .