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decoupled algorithm for transient viscoelastic flow modeling

KOREA-AUSTRALIA RHEOLOGY JOURNAL 2012年第1期24卷 53-63页

作者： Kwon, Youngdon Park, Kwang Sun Sungkyunkwan Univ Sch Chem Engn Suwon 440746 Gyeonggi Do South Korea

In the framework of finite element analysis we propose fast and robust time integration scheme for viscoelastic fluid (the Oldroyd-B and Leonov models) flow by way of efficient decoupling of equations. Developed algorithms of the 1st and 2nd order are shown to disclose convergence characteristics equivalent to conventional methods of corresponding order when applied to ID poiseuille and 2D creeping contraction flow problems. In comparison with fully coupled implicit technique, they notably enhance the computation speed. For the time dependent flow modeling with pressure difference imposed slightly below the steady limit, current as well as conventional approximation scheme has demonstrated fluctuating solution without approaching the steady state. From the result, we may conclude that the existence of upper limit for convergent steady solution implies flow transition to highly elastic time-fluctuating field without steady asymptotic. It is presumably associated with some real unstable elastic flow like re-entrant vortex oscillation and extrudate distortion outside the channel outlet.

关键词： decoupled algorithm viscoelastic flow finite element Leonov model contraction flow elastic instability

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STABILITY AND CONVERGENCE ANALYSIS OF A decoupled algorithm FOR A FLUID-FLUID INTERACTION PROBLEM

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2016年第5期54卷 2833-2867页

作者： Zhang, Yuhong Hou, Yanren Shan, Li Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Xi An Jiao Tong Univ Ctr Computat Geosci Xian 710049 Shaanxi Peoples R China Liaoning Tech Univ Coll Sci Fu Xin 123000 Peoples R China

In this paper, we analyze the stability and error estimate of a decoupled method for a fluid-fluid interaction problem. This method applies the partitioned time stepping method to decouple the coupling system, which is the implicit/explicit scheme and was first proposed by Connors, Howell, and Layton [SIAM T. Numer. Anal., 50 (2012), pp. 1297-1319]. Theoretically, we prove that this decoupled algorithm is not only conditional stable but also long time stable. We also derive that its error convergence order is o(Delta t + h(3/2)) for velocity variables in L-2 norm. Besides, several numerical tests are given to verify our theoretical results.

关键词： fluid-fluid interaction decoupled algorithm stability error convergence order

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Error estimates of a decoupled algorithm for a fluid-fluid interaction problem

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2018年 333卷 266-291页

作者： Zhang, Yuhong Hou, Yanren Shan, Li Hunan Normal Univ Coll Math & Comp Sci Key Lab High Performance Comp & Stochast Informat Changsha 410081 Hunan Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Xi An Jiao Tong Univ Ctr Computat Geosci Xian 710049 Shaanxi Peoples R China Liaoning Tech Univ Coll Sci Fu Xin 123000 Peoples R China

In this paper, we study the error estimates of a decoupled algorithm for the fluid-fluid model. The system consists of two Navier-Stokes equations which are coupled by a set of linear interface conditions. We apply the partitioned time stepping method to decouple the system. The corresponding scheme is unconditionally stable and we prove that the error estimates for velocities in L-2 norm are optimal. Moreover, under a restriction on the time step scale, we prove that the. convergent orders for the velocities in H-1 norm and for the pressures in L-2 norm are Delta t(7/8) + h and Delta t(3/4) + h, respectively. Two numerical examples are given to verify our theoretical results. Besides, by comparing the decoupled algorithm with the coupled one, numerical test shows the effectiveness of our decoupled method. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Fluid-fluid interaction decoupled algorithm Error estimates

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A second-order decoupled algorithm with different subdomain time steps for the non-stationary Stokes/Darcy model

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NUMERICAL algorithmS 2021年第3期88卷 1137-1182页

作者： Xue, Dandan Hou, Yanren Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China

In this paper, we propose and analyze a second-order decoupled algorithm with different subdomain time steps for the non-stationary Stokes/Darcy model. It is based on the second-order spectral deferred correction method in time and the finite element method in space. We provide the stability and convergence results of our decoupled scheme. Last, some numerical experiments are given to illustrate the accuracy and effectiveness of our decoupled scheme.

关键词： Stokes Darcy model decoupled algorithm Different time steps Spectral deferred correction

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New approach to prove the stability of a decoupled algorithm for a fluid-fluid interaction problem

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年 371卷

作者： Zhang, Yuhong Shan, Li Hou, Yanren Hunan Normal Univ Coll Math & Comp Sci Changsha 410081 Hunan Peoples R China Liaoning Tech Univ Coll Sci Fu Xin 123000 Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China Xi An Jiao Tong Univ Ctr Computat Geosci Xian 710049 Shaanxi Peoples R China

The decoupled algorithm for a fluid-fluid interaction problem was first proposed by Connors et al. (2012). Compared with the study on its long time stable which was presented in Zhang et al. (2016), we adopt a new approach to prove the stability of the decoupled method on a finite time interval (0, T] herein. By using the Brouwer's fixed point theorem, the existence of the numerical solution is proved. At the same time, by considering the error estimate between the exact solution and the numerical solution, we derive the stability of the decoupled method under the condition of Delta t + h <= c from the inductive method. Furthermore, we prove the uniqueness of the numerical solution when Delta t is appropriately small. The main theoretical results about the decoupled algorithm in this paper are new. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Fluid-fluid interaction decoupled algorithm Stability Brouwer's fixed point theorem

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NUMERICAL RECONSTRUCTION OF DIFFUSION AND POTENTIAL COEFFICIENTS FROM TWO OBSERVATIONS: decoupled RECOVERY AND ERROR ESTIMATES

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SIAM JOURNAL ON NUMERICAL ANALYSIS 2024年第5期62卷 2276-2307页

作者： Cen, Siyu Zhou, Zhi Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

The focus of this paper is on the concurrent reconstruction of both the diffusion and potential coefficients present in an elliptic/parabolic equation, utilizing two internal measurements of the solutions. A decoupled algorithm is constructed to sequentially recover these two parameters. In the first step, we implement a straightforward reformulation that results in a standard problem of identifying the diffusion coefficient. This coefficient is then numerically recovered, with no requirement for knowledge of the potential, by utilizing an output least-squares method coupled with finite element discretization. In the second step, the previously recovered diffusion coefficient is employed to reconstruct the potential coefficient, applying a method similar to the first step. Our approach is stimulated by a constructive conditional stability, and we provide rigorous a priori error estimates in L-2(Omega ) for the recovered diffusion and potential coefficients. To derive these estimates, we develop a weighted energy argument and suitable positivity conditions. These estimates offer a beneficial guide for choosing regularization parameters and discretization mesh sizes, in accordance with the noise level. Some numerical experiments are presented to demonstrate the accuracy of the numerical scheme and support our theoretical results.

关键词： identify multiple coefficients decoupled algorithm conditional stability least- squares approach finite element approximation error estimate

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Second order, fully decoupled, linear, exactly divergence-free and unconditionally stable discrete scheme for incompressible MHD equations

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2024年 169卷 195-204页

作者： Ding, Qianqian Mao, Shipeng Xi, Ruijie Beijing Jiaotong Univ Sch Math & Stat Beijing 100044 Peoples R China Univ Chinese Acad Sci Chinese Acad Sci Inst Computat Math & Sci Engn Comp Acad Math & Syst SciNCMISLSECSch Math Sci Beijing 100190 Peoples R China

This article designs a fully decoupled finite element algorithm with second order time-accuracy for the incompressible vector potential magnetohydrodynamic (MHD) system. The novel feature lies in the fact that it naturally produces an exactly divergence-free discretized solution of magnetic induction. The designed algorithm exhibits second-order accuracy, unconditional stability, linearity and fully decoupling. It is implemented by introducing the scalar auxiliary variable (SAV) techniques, combining second-order pressure-correction method, explicit treatment for the nonlinear/coupled terms, and a finite element method for spatial discretization. The effectiveness of this developed algorithm is demonstrated through various three-dimensional numerical simulations, including convergence tests and benchmark issues such as the driven cavity flow, the hydromagnetic Kelvin-Helmholtz instability and the island coalescence problem.

关键词： Vector potential MHD SAV decoupled algorithm Finite element method Crank-Nicolson scheme Unconditional stability

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A decoupled, unconditionally energy stable and charge-conservative finite element method for inductionless magnetohydrodynamic equations

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 2022年 127卷 80-96页

作者： Zhang, Xiaodi Ding, Qianqian Zhengzhou Univ Henan Acad Big Data Zhengzhou 450052 Peoples R China Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China Shandong Univ Sch Math Jinan 250100 Peoples R China

In this paper, we propose a decoupled, unconditionally energy stable and charge-conservative finite element method for the inductionless magnetohydrodynamic equations. The time marching is combined with a first order semi-implicit Euler scheme, a first order perturbation term and some delicate implicit-explicit treatments for coupling terms. The finite element discretization is based on mixed conforming elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, and the current density and electric potential are discretized by divergence-conforming face elements and discontinuous volume elements. This fully discrete scheme only needs to solve a linear subsystem at each time step and yields an exactly divergence -free current density directly. We show that the proposed scheme is well-posed and unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present the error estimates for velocity, current density and electric potential. Finally, the validity, reliability, and accuracy of the proposed algorithm are supported by several numerical experiments.

关键词： Inductionless MHD Conservation of charges decoupled algorithm Unconditional stability Error analysis Finite element method

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A decoupled and positivity-preserving discrete duality finite volume scheme for anisotropic diffusion problems on general polygonal meshes

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JOURNAL OF COMPUTATIONAL PHYSICS 2018年 372卷 773-798页

作者： Su, Shuai Dong, Qiannan Wu, Jiming China Acad Engn Phys Grad Sch Beijing 100088 Peoples R China Inst Appl Phys & Computat Math Beijing 100088 Peoples R China

We suggest a new positivity-preserving discrete duality finite volume (DDFV) scheme for anisotropic diffusion problems on general polygonal meshes. Like existing DDFV schemes, this scheme is built on the primary and dual meshes and has finite volume (FV) equations for both vertex and cell-centered unknowns. The transpose of the coefficient matrix for the FV equations of the cell-centered unknowns is an M-matrix while that for the vertex unknowns is not an M-matrix but a symmetric and positive definite matrix. By employing a certain truncation technique for the vertex unknowns, the positivity-preserving property for both categories of unknowns is guaranteed. Local conservation is strictly maintained for the cell-centered unknowns and conditionally maintained for the vertex unknowns. Since the FV equations of the vertex unknowns can be solved independently, the two sets of FV equations are decoupled. In contrast to existing nonlinear positivity-preserving schemes, the new scheme requires no nonlinear iterations for linear problems. For nonlinear problems, the positivity-preserving mechanism of the new scheme is decoupled from its nonlinear iteration so that any nonlinear solver can be adopted. Moreover, the positivity of the discrete solution is proved and the well-posedness is analyzed rigorously for linear problems. In numerical experiments, the new scheme is examined extensively and compared with two positivity-preserving cell-centered schemes and a nonlinear DDFV scheme. Numerical results show that the present DDFV scheme achieves second-order accuracy and preserves the positivity of the solution for heterogeneous and anisotropic problems on severely distorted grids. The high efficiency of the scheme is also demonstrated by the comparison of computation time and number of nonlinear iterations. (C) 2018 Elsevier Inc. All rights reserved.

关键词： Diffusion problems DDFV decoupled algorithm Positivity-preserving scheme Well-posedness

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decoupled two level finite element methods for the steady natural convection problem

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NUMERICAL algorithmS 2015年第4期68卷 837-866页

作者： Zhang, Tong Zhao, Xin Huang, Pengzhan Henan Polytech Univ Sch Math & Informat Sci Jiaozuo 454003 Peoples R China Baoji Univ Arts & Sci Dept Geog Sci & Environm Engn Baoji 721013 Peoples R China Xinjiang Univ Coll Math & Syst Sci Urumqi 830046 Peoples R China

In this work three kinds of decoupled two level finite element methods are proposed and analyzed for the natural convection problem. Firstly, some a priori bounds and the optimal error estimates of velocity and temperature in L (2) norm are provided for the standard Galerkin finite element method. Secondly, by using the coarse grid numerical solutions to decouple the nonlinear coupling term, we establish the convergence results for the proposed decoupled two level finite element schemes with meshes h and H satisfy h=H (2). Finally, two numerical examples are presented to show the efficiency and effectiveness of the proposed algorithms for the steady natural convection problem.

关键词： Natural convection problem decoupled algorithm Two level method Linearization

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