A Defect-Deferred Correction Method for Fluid-Fluid Interaction

Abstract

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 projection based variational multiscale method for a fluid–fluid interaction problem

“…Herein, for simplicity pressures are set to zero in both domains, and right hand side forcing, boundary and two initial values are computed using the manufactured true solution as is done in [1]. Problem parameters, b = 1/2, κ = 0.001 and the final time T = 1 are fixed while a, ν 1 and ν 2 vary from one computation to the other.…”

“…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.…”

“…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 order accuracy in temporal discretization.…”

Stability of Two Conservative, High-Order Fluid-Fluid Coupling Methods

“…Previous work has shown that coupling algorithms can be designed with unconditional stability, but these methods were low-order accurate and not conservative [5,6]. More recently, it has been shown in [1] that higher order can be achieved using deferred correction techniques, but these methods require multiple invocations of each module at each time step, using different equations for the internal dynamics. Furthermore, they have not yet incorporated conservation and multirate time stepping.…”

Higher Accuracy Methods for Fluid Flows in Various Applications: Theory and Implementation

“…In order to show the improved accuracy for the correction approximation, we will need the following result. In order to keep this chapter from being prohibitively long, the proof is given in full detail in [23]. Consider Let…”