A model of two incompressible, Newtonian fluids coupled across a common interface is studied. The nonlinearity of the coupling condition exacerbates the problem of decoupling the fluid calculations in each subdomain, a natural parallelization strategy employed in current climate models. A specialized partitioned time stepping method is studied which decouples the discrete fluid equations without sacrificing stability and maintaining convergence. This is accomplished through explicit updating of the size of the jump in tangential velocities across the fluid-fluid interface by a geometric averaging of this data over the previous two time levels. A full numerical analysis is presented and computational tests are performed demonstrating the robustness of this method.
We consider multiphysics applications from algorithmic and architectural perspectives, where “algorithmic” includes both mathematical analysis and computational complexity, and “architectural” includes both software and hardware environments. Many diverse multiphysics applications can be reduced, en route to their computational simulation, to a common algebraic coupling paradigm. Mathematical analysis of multiphysics coupling in this form is not always practical for realistic applications, but model problems representative of applications discussed herein can provide insight. A variety of software frameworks for multiphysics applications have been constructed and refined within disciplinary communities and executed on leading-edge computer systems. We examine several of these, expose some commonalities among them, and attempt to extrapolate best practices to future systems. From our study, we summarize challenges and forecast opportunities.
Abstract. There have been many numerical simulations but few analytical results of stability and accuracy of algorithms for computational modeling of fluid-fluid and fluid-structure interaction problems, where two domains corresponding to different fluids (ocean-atmosphere) or a fluid and deformable solid (blood flow) are separated by an interface. As a simplified model of the first examples, this report considers two heat equations in Ω 1 , Ω 2 ⊂ R 2 adjoined by an interface I = Ω 1 ∩ Ω 2 ⊂ R. The heat equations are coupled by a condition that allows energy to pass back and forth across the interface I while preserving the total global energy of the monolithic, coupled problem. To compute approximate solutions to the above problem only using subdomain solvers, two first order in time, fully discrete methods are presented. The methods consist of an implicit-explicit (IMEX) approach, in which the action across I is lagged and a partitioned method based on passing interface values back and forth across I. Stability and convergence results are derived for both schemes. Numerical experiments that support the theoretical results are presented.
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...
The estimation of discretization error in numerical simulations is a key component in the development of uncertainty quantification. In particular, there exists a need for reliable, robust estimators for finite volume and finite difference discretizations of hyperbolic partial differential equations. The approach espoused here, often called the error transport approach in the literature, is to solve an auxiliary error equation concurrently with the primal governing equation to obtain a point-wise (cell-wise) estimate of the discretization error. One-dimensional, nonlinear, time-dependent problems are considered. In contrast to previous work, fully nonlinear error equations are advanced, and potential benefits are identified. A systematic approach to approximate the local residual for both method-of-lines and space-time discretizations is developed. Behavior of the error estimates on problems that include weak solutions demonstrates that this nonlinear error evolution provides useful error estimates.
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