A defect-deferred correction method, increasing both temporal and spatial accuracy, for fluid-fluid interaction problem with nonlinear interface condition is considered by geometric averaging of the previous two-time levels. In the defect step, an artificial viscosity is added only on the fluctuations in the velocity gradient by removing this effect on a coarse mesh. The dissipative influence of the artificial viscosity is further eliminated in the correction step while gaining additional temporal accuracy at the same time. The stability and accuracy analyses of the resulting algorithm are investigated both analytically and numerically.Keywords: fluid-fluid interaction, subgrid artificial viscosity, defect-deferred correction Here, the domain Ω ⊂ R d , (d = 2, 3) is a polygonal or polyhedral domain that consists of two subdomains Ω 1 and Ω 2 , coupled across an interface I, for times t ∈ [0, T ]. The unknown velocity fields and pressure are denoted by u i and p i . Also, | · | represents the Euclidean norm and the vectorŝ n i represent the unit normals on ∂Ω i , and τ is any vector such that τ ·n i = 0. Further, the kinematic viscosity is ν i and the body forcing on velocity field is f i in each subdomain. Here, κ denote the friction parameter for which frictional drag force is assumed to be proportional to the square of the jump of the velocities across the interface.The main characteristic of the proposed defect-deferred correction (DDC) algorithm is the use of a projection-based variational multiscale method (VMS) as a predictor (defect) step for fluid-fluid interaction problems. Here, the geometric averaging (GA) of the coupling terms is considered at the interface. In VMS, since stabilization acts only on the fluctuations in the velocity gradient, the proposed algorithm is called subgrid artificial viscosity (SAV) based defect-deferred correction (SAV-DDC) method. New SAV based defect step indeed increases the efficiency of the DDC method. The scheme replaces the artificial viscosity (AV) step of the defect-deferred correction (AV-DDC) * method of [3] by the SAV step. For smooth solutions, Theorem 5.3 shows the error of the SAV-DDC algorithm is second order in time. Section 6 includes numerical tests to confirm theory and establishes the advantages of the proposed approach over AV-DDC. Related WorksIn recent years, the atmosphere-ocean interaction problem has been attracted by many scientists to contribute to the simulation of these complex flows. For example, Refs. [4,5,6,7] studied modeling of atmosphere-ocean problems and their numerical analysis. Different treatments of coupling terms at the interface are derived to improve the solution of these problems. The method in [8] uses nonlinear interface conditions, whereas, in [9], interface conditions for two heat equations are linearly coupled. In [10], a decoupling approach, known as GA of the coupling terms at the interface, is introduced for nonlinear coupling of two Navier-Stokes equations. This idea leads to a decoupled and unconditionally stable method...

A method is presented, that combines the defect and deferred correction approaches to approximate solutions of Navier–Stokes equations at high Reynolds number. The method is of high accuracy in both space and time, and it allows for the usage of legacy codes a frequent requirement in the simulation of turbulent flows in complex geometries. The two‐step method is considered here; to obtain a regularization that is second order accurate in space and time, the method computes a low‐order accurate, stable, and computationally inexpensive approximation (Backward Euler with artificial viscosity) twice. The results are readily extendable to the higher order accuracy cases by adding more correction steps. Both the theoretical results and the numerical tests provided demonstrate that the computed solution is stable and the accuracy in both space and time is improved after the correction step. We also perform a qualitative test to demonstrate that the method is capable of capturing qualitative features of a turbulent flow, even on a very coarse mesh. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 814–839, 2017

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...

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