This paper extends proper orthogonal decomposition reduced order modeling to flows governed by double diffusive convection, which models flow driven by two potentials with different rates of diffusion. We propose a reduced model based on proper orthogonal decomposition, present a stability and convergence analyses for it, and give results for numerical tests on a benchmark problem which show it is an effective approach to model reduction in this setting.

In this paper, we propose, analyze and test a post-processing implementation of a projectionbased variational multiscale (VMS) method with proper orthogonal decomposition (POD) for the incompressible Navier-Stokes equations. The projection-based VMS stabilization is added as a separate post-processing step to the standard POD approximation, and since the stabilization step is completely decoupled, the method can easily be incorporated into existing codes, and stabilization parameters can be tuned independent from the time evolution step. We present a theoretical analysis of the method, and give results for several numerical tests on benchmark problems which both illustrate the theory and show the proposed method's effectiveness.

The proposed method aims to approximate a solution of a fluid-fluid interaction problem in case of low viscosities. The nonlinear interface condition on the joint boundary allows for this problem to be viewed as a simplified version of the atmosphere-ocean coupling. Thus, the proposed method should be viewed as potentially applicable to air-sea coupled flows in turbulent regime. The method consists of two key ingredients. The geometric averaging approach is used for efficient and stable decoupling of the problem, which would allow for the usage of preexisting codes for the air and sea domain separately, as "black boxes". This is combined with the variational multiscale stabilization technique for treating flows at high Reynolds numbers. We prove the stability and accuracy of the method, and provide several numerical tests to assess both the quantitative and qualitative features of the computed solution.Numerical methods for solving this type of coupled problems in laminar flow regime have been investigated [3,6,27,1]. In [6], IMEX and geometric averaging (GA) time stepping methods have been proposed (and further developed in [1]) for the Navier-Stokes equations with nonlinear interface condition.The study of AO interaction has received considerable interest in the last thirty years, starting with the seminal paper of Lions, Temam and Wang, [21,22], on the analysis of full equations for AO flow. Today, many models exist and an abundance of software code is available for climate models (both global and regional), hurricane propagation, coastal weather prediction, etc. see, e.g., [2,4,25] and references therein. The reasoning behind most of these models is as follows: the boundary condition on the joint AO interface must be chosen in such a way, that fluxes of conserved quantities are allowed to pass from one domain to the other. In particular, the nonlinear interface condition (1.2), together with (1.3) ensures that the energy is being passed between the two domains in the model above, with the global energy still being conserved.The AO coupling problem (as well as its modest version, the fluid-fluid interaction with nonlinear coupling, considered in this report) provides many challenges. In addition to the usual issues one has to overcome when solving the Navier-Stokes equations, the AO models should allow to use different spacial and temporal scales for the atmosphere and ocean domains, as the energy in the atmosphere remains significant at smaller time scales and larger spatial scales, than the energy of the ocean. In order to do so, as well as make use of the existing codes written separately for the fluid flows in the air or the ocean domains, one needs to create partitioned methods, that allow for a stable and accurate decoupling of the AO system.The literature on numerical analysis of time-dependent coupling problem (1.1)-(1.6) is somewhat scarse; some approaches to creating a stable, accurate, computationally attractive decoupling method can be found in [3,6,5,20,27,1]. The methods in [20, 1] provide second...

The present work is devoted to introduce the backward Euler based modular time filter method for MHD flow. The proposed method improves the accuracy of the solution without a significant change in the complexity of the system. Since time filters for fluid variables are added as separate post processing steps, the method can be easily incorporated into an existing backward Euler code. We investigate the conservation and long time stability properties of the improved scheme. Stability and second order convergence of the method are also proven. The influences of introduced time filter method on several numerical experiments are given, which both verify the theoretical findings and illustrate its usefulness on practical problems.Here, u, P := p + s 2 |B| 2 , p and B denote the unknown velocity, modified pressure, pressure and magnetic field, respectively. The body forces f and ∇ × g are forcing on the velocity and magnetic field, respectively. Also, Re is the Reynolds number, Re m is the magnetic Reynolds number, and s is the coupling number. The Lagrange multiplier (dummy variable) λ corresponds to the solenoidal constraint on the magnetic field. In the continuous case, provided the initial condition B 0 is solenoidal, then the use of λ is unnecessary, see [5]. However, when discretizing with the finite element method, this solenoidal constraint is needed to be enforced explicitly and thus the additional variable is required. We also assume that the system (1.1)-(1.4) is equipped with homogeneous Dirichlet boundary conditions for velocity and the magnetic field.Due to the coupling of the equations of the velocity and the magnetic field, developing efficient, accurate numerical methods for solving MHD system (1.1)-(1.4) remains a great challenge in computational fluid dynamics community. It is well known that time filter methods combined with leapfrog scheme are commonly used in geophysical fluid dynamics to reduce spurious oscillations to improve predictions, see e.g. [3], but these methods degrade the numerical accuracy and over damps the physical mode. A successfully tuned model was developed by Williams [29] reducing undesired numerical damping of [3] with higher order accuracy, see [2,22,24,30] and references therein. On the other hand, in practice, the use of the backward Euler method is often preferred to extend a code for the steady state problem and this yields stable but inefficient time accurate transient solutions, see [10]. To improve this behavior, time filters are used to stabilize the backward Euler discretizations in [14] for the classical numerical ODE theory.The present work extends the method of [7] tailored to MHD flows for constant time step. As it is mentioned in this study, the constant time step method is equivalent to a general second order, two step and A-stable method given in [9] and [19]. The scheme we consider is the time filtered backward Euler method, which is efficient, O(∆t 2 ) and amenable to implementation in existing legacy codes. In addition, we also consider the numerical conse...