Second-order Time-accurate Scheme Research Articles

A fully-decoupled energy stable scheme for the phase-field model of non-Newtonian two-phase flows

<abstract><p>In this paper, we first propose a novel fully-decoupled, linear and second-order time accurate scheme to solve the phase-field model of non-Newtonian two-phase flows; the developed scheme is based on a stabilized Scalar Auxiliary Variable (SAV) approach. We strictly prove the unconditional energy stability of the scheme and conduct a numerical simulation to show the accuracy and stability of the proposed scheme. Moreover, we can observe that the parameter $ r $ in non-Newtonian fluids can affect spatial patterns during phase transitions, which directly enables us to design and perform optimal control experiments in engineering processes.</p></abstract>

A novel hybrid IGA-EIEQ numerical method for the Allen–Cahn/Cahn–Hilliard equations on complex curved surfaces

We present an efficient fully discrete algorithm for solving the Allen–Cahn and Cahn–Hilliard equations on complex curved surfaces. The spatial discretization employs the recently developed IGA (isogeometric analysis) framework, where we adopt the strategy of Loop subdivision with the superior adaptability of any topological structure, and the basis functions are quartic box-splines used to define the subdivided surface. The time discretization is based on the so-called EIEQ (explicit-Invariant Energy Quadratization) approach, which applies multiple newly defined variables to linearize the nonlinear potential and realize the efficient decoupled type computation. The combination of these two methods can help us to gain a linear, second-order time accurate scheme with the property of unconditional energy stability, whose rigorous proof is given. We also develop a nonlocal splitting technique such that we only need to solve decoupled, constant-coefficient elliptic equations at each time step. Finally, the effectiveness of the developed numerical algorithm is verified by various numerical experiments on the complex benchmark curved surfaces such as bunny, splayed, and head surfaces.

Fully-Decoupled and Second-Order Time-Accurate Scheme for the Cahn–Hilliard Ohta–Kawaski Phase-Field Model of Diblock Copolymer Melt Confined in Hele–Shaw Cell

Fully-Decoupled and Second-Order Time-Accurate Scheme for the Cahn–Hilliard Ohta–Kawaski Phase-Field Model of Diblock Copolymer Melt Confined in Hele–Shaw Cell

Original variables based energy-stable time-dependent auxiliary variable method for the incompressible Navier–Stokes equation

In this study, we develop an efficiently linear and energy-stable method for the incompressible Navier–Stokes equation. A time-dependent Lagrange multiplier is introduced to change the original equation into an equivalent form. Using the equivalent equation, we design a second-order time-accurate scheme based on the second-order backward difference formula (BDF2). The proposed scheme explicitly treats the advection term. In each time iteration, some linear elliptic type equations need to be solved. Therefore, the calculation is highly efficient. Moreover, the time-discretized energy stability with respect to original variables can be easily proved. Various benchmark tests, such as lid-driven cavity flow, Kelvin–Helmholtz instability, and Taylor–Green vortices, are performed to validate the performance of the proposed method.

Efficient Fully decoupled and second-order time-accurate scheme for the Navier–Stokes coupled Cahn–Hilliard Ohta–Kawaski Phase-Field model of Diblock copolymer melt

In this work, we aim to develop a highly efficient numerical scheme for the flow-coupled phase-field model of diblock copolymer melt. Formally, the model is a very complicated nonlinear system that consists of the Navier–Stokes equations and the Cahn–Hilliard type equations with the Ohta–Kawaski potential. Through a combination of a novel decoupling technique and the projection method, we develop the first full decoupling, energy stable, and second-order time-accurate numerical scheme for this model. The decoupling technique is based on the design of an auxiliary ODE, which plays a vital role in obtaining the full decoupling structure while maintaining energy stability. The high efficiency of the scheme is not only reflected by its linear and decoupled structure but also because it only needs to solve a few elliptic equations at each time step. We strictly prove that the scheme satisfies the unconditional energy stability and give a detailed implementation process. Numerical experiments further verify the convergence rate, energy stability, and effectiveness of the developed algorithm.

A new efficient fully-decoupled and second-order time-accurate scheme for Cahn–Hilliard phase-field model of three-phase incompressible flow

For the highly coupled and nonlinear Cahn–Hilliard phase-field model of three-phase incompressible flow, how to establish a fully-decoupled numerical scheme with second-order time accuracy has always been a very difficult and unsolved problem. In this paper, we propose a novel decoupling method, which only needs to solve several decoupling linear elliptic equations with constant coefficients at each time step to obtain a numerical solution with second-order time accuracy. The key idea is to introduce two nonlocal auxiliary variables into the system, one of which is used to linearize the nonlinear potential, and the other is used to introduce an ordinary differential equation to deal with the nonlinear coupling terms with “zero-energy-contribution” characteristics. We strictly prove the solvability and unconditional energy stability of the scheme, and conduct numerical simulations in 2D and 3D to show the accuracy and stability of the scheme numerically. To the best of the author’s knowledge, the method developed in this paper is the first second-order fully-decoupled scheme for the hydrodynamics coupled phase-field model.

A novel fully-decoupled, second-order and energy stable numerical scheme of the conserved Allen–Cahn type flow-coupled binary surfactant model

In this paper, we establish a binary fluid surfactant model by coupling two mass-conserved Allen–Cahn equations and the Navier–Stokes equations and consider numerical simulations of the developed model. Due to a large number of nonlinear and nonlocal coupling terms in the model, it is very challenging to design an efficient and accurate numerical scheme, especially the full decoupling scheme with second-order time accuracy. We solve this challenge by developing a novel fully-decoupled approach, where the key idea achieving the full decoupling structure is to introduce an ordinary differential equation to deal with the nonlinear coupling terms that satisfy the so-called “zero-energy-contribution” property. In this way, we can easily discretize the coupled nonlinear terms in a fully explicit way, while still maintaining unconditional energy stability. By combining with the projection type method and quadratization approach, at each time step, we only need to solve several fully-decoupled linear elliptic equations with constant coefficients. We strictly prove the solvability of the scheme, prove that the scheme satisfies the unconditional energy stability, and give various 2D and 3D numerical simulations to show its stability and accuracy numerically. As far as the author knows, this is the first fully-decoupled and second-order time-accurate scheme of the flow-coupled phase-field type model.

An unconditionally energy-stable second-order time-accurate scheme for the Cahn–Hilliard equation on surfaces

In this paper, we propose an unconditionally energy-stable second-order time-accurate scheme for the Cahn–Hilliard equation on surfaces. The discretization is performed via a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation. The proposed scheme, which combines a Crank–Nicolson-type scheme with a linearly stabilized splitting scheme, is second-order accurate in time. The discrete system is shown to be conservative and unconditionally energy-stable. The resulting system of discrete equations is simple to implement, and can be solved using a biconjugate gradient stabilized method. We demonstrate the performance of our proposed algorithm through several numerical experiments.

A stable second-order scheme for fluid–structure interaction with strong added-mass effects

In this paper, we present a stable second-order time accurate scheme for solving fluid–structure interaction problems. The scheme uses so-called Combined Field with Explicit Interface (CFEI) advancing formulation based on the Arbitrary Lagrangian–Eulerian approach with finite element procedure. Although loosely-coupled partitioned schemes are often popular choices for simulating FSI problems, these schemes may suffer from inherent instability at low structure to fluid density ratios. We show that our second-order scheme is stable for any mass density ratio and hence is able to handle strong added-mass effects. Energy-based stability proof relies heavily on the connections among extrapolation formula, trapezoidal scheme for second-order equation, and backward difference method for first-order equation.Numerical accuracy and stability of the scheme is assessed with the aid of two-dimensional fluid–structure interaction problems of increasing complexity. We confirm second-order temporal accuracy by numerical experiments on an elastic semi-circular cylinder problem. We verify the accuracy of coupled solutions with respect to the benchmark solutions of a cylinder-elastic bar and the Navier–Stokes flow system. To study the stability of the proposed scheme for strong added-mass effects, we present new results using the combined field formulation for flexible flapping motion of a thin-membrane structure with low mass ratio and strong added-mass effects in a uniform axial flow. Using a systematic series of fluid–structure simulations, a detailed analysis of the coupled response as a function of mass ratio for the case of very low bending rigidity has been presented.

An analysis of a second-order time accurate scheme for a finite volume method for parabolic equations on general nonconforming multidimensional spatial meshes

The present work is an extension of our previous work (Bradji and Fuhrmann (2011) [4]) which dealt with error analysis of a finite volume scheme of first order (both in time and space) for parabolic equations on general nonconforming multidimensional spatial meshes introduced recently in Eymard et al. (2010) [12]. We aim in this paper to get a higher-order time accurate scheme for a finite volume method for parabolic equations using the same class of spatial generic meshes stated above.We derive a finite volume scheme approximating the heat equation, as a model for parabolic equations, in which the discretization in time is performed using the Crank–Nicolson method.We derive an a priori estimate for the discrete problem and we prove that the error estimate of the finite volume scheme is of order two in time and it is of optimal order in space.The error estimate is analysed in several norms which allow us to derive approximations for the exact solution and its first derivatives (both spatial and temporal) whose the convergence order is two in time and it is optimal in space.We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, that the convergence order is k2+hD, where hD (resp. k) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption u∈C3([0,T];C2(Ω¯)) for the exact solution u.The proof of these error estimates is based essentially on a new a priori estimate for the discrete problem and a comparison between the finite volume approximate solution and an auxiliary finite volume approximation.

A second-order time accurate finite element method for quasi-incompressible viscous flows

A finite element method for quasi-incompressible viscous flows is presented. An equation for pressure is derived from a second-order time accurate Taylor–Galerkin procedure that combines the mass and the momentum conservation laws. At each time step, once the pressure has been determined, the velocity field is computed solving discretized equations obtained from another second-order time accurate scheme and a least-squares minimization of spatial momentum residuals. The terms that stabilize the finite element method (controlling wiggles and circumventing the Babuska–Brezzi condition) arise naturally from the process, rather than being introduced a priori in the variational formulation. A comparison between the present second-order accurate method and our previous first-order accurate formulation is shown. The method is also demonstrated in the computation of the leaky-lid driven cavity flow and in the simulation of a crossflow past a circular cylinder. In both cases, good agreement with previously published experimental and computational results has been obtained. Copyright © 2010 John Wiley & Sons, Ltd.

Transition vers la turbulence géostrophique pour un écoulement d'air en cavité tournante différentiellement chauffée

A direct numerical simulation is carried out to determine the transition to geostrophic turbulence in an air-filled differentially heated rotating annulus. The coupled Navier–Stokes and energy equations are solved numerically with a spectral method based on collocation Chebyshev and Fourier approximations associated with a second-order time accurate scheme. For a temperature difference fixed at ΔT=5 K, when increasing the rotation rate, we have obtained the successive occurrence of the different regimes as described in the literature. The route to turbulent flow, studied with respect to two principal dimensionless parameters, the Taylor number Ta ∗ and the thermal Rossby number Θ, presents steady axisymmetric solution ( S), periodic flow ( P 1), quasi-periodic flow ( QP), periodic flow ( Pn), before transition to chaotic motion ( NP). To cite this article: P. Maubert, A. Randriamampianina, C. R. Mecanique 330 (2002) 365–370.