arXiv subjects

Xiaoming Wang

Publications and source records attributed to Xiaoming Wang.

78 records · Page 5Linked to original sources

arXiv2014

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 which treat the nonlinear term explicitly (backward differentiation formula with a one-leg method). 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.

math.NA↗

arXiv2012

Efficient and Long-Time Accurate Second-Order Methods for Stokes-Darcy System

We propose and study two second-order in time implicit-explicit (IMEX) methods for the coupled Stokes-Darcy system that governs flows in karst aquifers. The first is a combination of a second-order backward differentiation formula and the second-order Gear's extrapolation approach. The second is a combination of the second-order Adams-Moulton and second-order Adams-Bashforth methods. Both algorithms only require the solution of two decoupled problems at each time step, one Stokes and the other Darcy. Hence, these schemes are very efficient and can be easily implemented using legacy codes. We establish the unconditional and uniform in time stability for both schemes. The uniform in time stability leads to uniform in time control of the error which is highly desirable for modeling physical processes, e.g., contaminant sequestration and release, that occur over very long time scales. Error estimates for fully-discretized schemes using finite element spatial discretizations are derived. Numerical examples are provided that illustrate the accuracy, efficiency, and long-time stability of the two schemes.

math.NA↗

arXiv2011

An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations

We investigate the long tim behavior of the following efficient second order in time scheme for the 2D Navier-Stokes equation in a periodic box: $$ \frac{3ω^{n+1}-4ω^n+ω^{n-1}}{2k} + \nabla^\perp(2ψ^n-ψ^{n-1})\cdot\nabla(2ω^n-ω^{n-1}) - νΔω^{n+1} = f^{n+1}, \quad -Δψ^n = \om^n. $$ The scheme is a combination of a 2nd order in time backward-differentiation (BDF) and a special explicit Adams-Bashforth treatment of the advection term. Therefore only a linear constant coefficient Poisson type problem needs to be solved at each time step. We prove uniform in time bounds on this scheme in $\dL2$, $\dH1$ and $\dot{H}^2_{per}$ provided that the time-step is sufficiently small. These time uniform estimates further lead to the convergence of long time statistics (stationary statistical properties) of the scheme to that of the NSE itself at vanishing time-step. Fully discrete schemes with either Galerkin Fourier or collocation Fourier spectral method are also discussed.

math.NA↗

arXiv2011

Long time stability of a classical efficient scheme for two dimensional Navier-Stokes equations

We prove that a popular classical implicit-explicit scheme for the 2D incompressible Navier--Stokes equations that treats the viscous term implicitly while the nonlinear advection term explicitly is long time stable provided that the time step is sufficiently small in the case with periodic boundary conditions. The long time stability in the $L^2$ and $H^1$ norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the NSE itself at vanishing time step. Both semi-discrete in time and fully discrete schemes with either Galerkin Fourier spectral or collocation Fourier spectral methods are considered.

math.NA↗

arXiv2010

Well-posedness of the Hele-Shaw-Cahn-Hilliard system

We study the well-posedness of the Hele-Shaw-Cahn-Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in $H^s, s>\frac{d}{2}+1$, the existence and uniqueness of solution in $C([0, T]; H^s)\cap L^2(0, T; H^{s+2})$ that is global in time in the two dimensional case ($d=2$) and local in time in the three dimensional case ($d=3$) are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood-Paley theory in order to establish certain key commutator estimates.

math.AP↗

arXiv2010

Partial and spectral-viscosity models for geophysical flows

Two models based on the hydrostatic primitive equa- tions are proposed. The first model is the primitive equations with partial viscosity only, and is oriented towards large-scale wave structures in the ocean and atmosphere. The second model is the viscous primitive equations with spectral eddy viscosity, and is oriented towards turbulent geophysical flows. For both models, the existence and uniqueness of global strong solutions is estab- lished. For the second model, the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zero is also established.

math.AP↗