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A unified implicit scheme for kinetic model equations. Part I. Memory reduction technique

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Science Bulletin 2017年第2期62卷 119-129页

作者： Songze Chen Chuang Zhang Lianhua Zhu Zhaoli Guo State Key Laboratory of Coal Combustion School of Energy and Power Engineering Huazhong University of Science and Technology Wuhan 430074 China

A memory reduction technique is proposed for solving stationary kinetic model equations. As implied by an integral solution of the stationary kinetic equation, a velocity distribution function can be reconstructed from given macroscopic variables. Based on this fact, we propose a technique to reconstruct distribution function at discrete level, and employ it to develop an implicit numerical method for kinetic equations. The new implicit method only stores the macroscopic quantities which appear in the collision term, and does not store the distribution functions. As a result, enormous memory requirement for solving kinetic equations is totally relieved. Several boundary conditions, such as, inlet, outlet and isothermal boundaries, are discussed. Some numerical tests demonstrate the validity and efficiency of the *** new implicit solver provides nearly identical solution as the explicit kinetic solver, while the memory requirement is on the same order as the Navier–Stokes solver.

关键词： implicit scheme Kinetic equation Memory reduction

AN　implicit　scheme　FOR　THE　NUMERICAL　SOLUTION　OF　THE　TIMEDEPENDENT　HEAD／DISK　DYNAMICS　IN　HARD　DISK　DRIVER　BY　FINITE　ELEMENT　METHOD

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Journal of Hydrodynamics 1994年第2期6卷 23-31页

作者： Hua Wei Tan Bing-lin He You-sheng(Shanghai Jiao Tong University, Shanghai, 200030, P. R. China )

An implicit scheme discretized by the finite element method is developed for the numerical solution of the time-dependent head/disk dynamics. The scheme is second-order timeaccurate, which offers the flexibility of employing time steps compatible with the system dynamics while minimizing the numerical instability presented in most explicit methods. Good agreement is orreryed between the present calculations and D. K. Miu's work.

关键词： implicit scheme finite element method time-dependent flying characteristics of slider

Analysis of an implicit Finite Difference scheme for Time Fractional Diffusion Equation

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Chinese Quarterly Journal of Mathematics 2016年第1期31卷 69-81页

作者： MA Yan College of Technology and Engineering Lanzhou University of Technology

Time fractional diffusion equation is usually used to describe the problems involving non-Markovian random walks. This kind of equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α∈(0, 1). In this paper, an implicit finite difference scheme for solving the time fractional diffusion equation with source term is presented and analyzed, where the fractional derivative is described in the Caputo sense. Stability and convergence of this scheme are rigorously established by a Fourier analysis. And using numerical experiments illustrates the accuracy and effectiveness of the scheme mentioned in this paper.

关键词： time fractional diffusion equation finite difference approximation implicit scheme stability convergence effectiveness

implicit discontinuous Galerkin method on agglomerated high-order grids for 3D simulations

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Chinese Journal of Aeronautics 2016年第6期29卷 1496-1505页

作者： Qin Wanglong Lyu Hongqiang Wu Yizhao Zhou Shijie Chen Zhengwu Department of Aerodynamics Nanjing University of Aeronautics and AstronauticsNanjing 210016China China Aerodynamics Research and Development Center Mianyang 621000China

High quality of geometry representation is regarded essential for high-order methods to maintain their high-order accuracy. An agglomerated high-order mesh generating method is investigated in combination with discontinuous Galerkin(DG) method for solving the 3D compressible Euler and Navier-Stokes equations. In this method, a fine linear mesh is first generated by standard commercial mesh generation tools. By taking advantage of an agglomeration method, a quadratic high-order mesh is quickly obtained, which is coarse but provides a high-quality geometry representation, thus very suitable for high-order computations. High-order discretizations are performed on the obtained grids with DG method and the discretized system is treated fully implicitly to obtain steady state solutions. Numerical experiments on several flow problems indicate that the agglomerated high-order mesh works well with DG method in dealing with flow problems of curved geometries. It is also found that with a fully implicit discretized system and a p-sequencing method, the DG method can achieve convergence state within several time steps which shows significant efficiency improvements compared to its explicit counterparts.

关键词： Agglomeration Discontinuous Galerkin(DG) High-order implicit scheme Navier-Stokes equations

A Gas-Kinetic Unified Algorithm for Non-Equilibrium Polyatomic Gas Flows Covering Various Flow Regimes

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Communications in Computational Physics 2021年第6期30卷 144-189页

作者： Wen-Qiang Hu Zhi-Hui Li Ao-Ping Peng Xin-Yu Jiang National Laboratory for Computational Fluid Dynamics Beijing University of Aeronautics and AstronauticsBeijing 100191China Hypervelocity Aerodynamics Institute China Aerodynamics Research and Development CenterMianyang 621000China

In this paper,a gas-kinetic unified algorithm(GKUA)is developed to investigate the non-equilibrium polyatomic gas flows covering various *** on the ellipsoidal statistical model with rotational energy excitation,the computable modelling equation is presented by unifying expressions on the molecular collision relaxing parameter and the local equilibrium distribution *** constructing the corresponding conservative discrete velocity ordinate method for this model,the conservative properties during the collision procedure are preserved at the discrete level by the numerical method,decreasing the computational storage and *** and implicit lower-upper symmetric Gauss-Seidel schemes are constructed to solve the discrete hyperbolic conservation equations *** the new GKUA,some numerical examples are simulated,including the Sod Riemann problem,homogeneous flow rotational relaxation,normal shock structure,Fourier and Couette flows,supersonic flows past a circular cylinder,and hypersonic flow around a plate placed *** results obtained by the analytic,experimental,direct simulation Monte Carlo method,and other measurements in references are compared with the GKUA results,which are in good agreement,demonstrating the high accuracy of the present ***,some polyatomic gas non-equilibrium phenomena are observed and analysed by solving the Boltzmann-type velocity distribution function equation covering various flow regimes.

关键词： Gas-kinetic unified algorithm polyatomic gas ellipsoidal statistical model conservative discrete velocity ordinate method implicit scheme

Numerical Simulations of Rarefied Gases in Curved Channels: Thermal Creep, Circulating Flow, and Pumping Effect

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Communications in Computational Physics 2009年第10期6卷 919-954页

作者： Kazuo Aoki Pierre Degond Luc Mieussens Department of Mechanical Engineering and Science Graduate School of EngineeringKyoto UniversityKyoto 606-8501Japan Universite de Toulouse UPSINSAUT1UTMInstitut de Mathematiques de ToulouseF-31062 ToulouseFrance CNRS Institut de Mathematiques de Toulouse UMR 5219F-31062 ToulouseFrance Universite de Bordeaux Institut de Mathematiques de Bordeaux351cours de la Liberation33405 Talence cedexFrance

We present numerical simulations of a new system of micro-pump based on the thermal creep effect described by the kinetic theory of *** device is made of a simple smooth and curved channel with a periodic temperature *** the Boltzmann-BGK model as the governing equation for the gas flow,we develop a numerical method based on a deterministic finite volume scheme,implicit in time,with an implicit treatment of the boundary *** method is comparatively less sensitive to the slow flow velocity than the usual Direct Simulation Monte Carlo method in case of long devices,and turns out to be accurate enough to compute macroscopic quantities like the pressure field in the *** simulations show the ability of the device to produce a one-way flow that has a pumping effect.

关键词： Knudsen compressor thermal creep flow transpiration flow BGK model implicit scheme

An implicit method using contravariant velocity components and its application to calculations in a harbour-channel area

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Acta Oceanologica Sinica 1998年第4期17卷 423-432页

作者： Shi Fengyan, Kong Yazhen and Ding Pingxing (State Key Laboratory of Estuarine and Coastal Research, East China Normal University. Shanghai 200062, China) State Key Lab. Estuarine Coast. Res. East China Normal University Shanghai 200062 China

The key problem in the computation of fluid dynamics using fine boundary-fitted grids is how to improve the numerical stability and decrease the calculating quantity. To solve this problem, implicit schemes should be adopted since explicit schemes may bring about a great increase in computation quantity according to the Courant-FrledrichsLewy condition. Whereas the adoption of implicit schemes is difficult to be realized because of the existence of two partial derivatives of surface elevations with respect to variables of alternative direction coordinates in each momentum equation in non-rectangular coordinates. With an aim to design an implicit scheme in non-reetangular ccordinates in the present paper, new momentum equations with the contravariant components of velocity vector are derived based on the shallow water dynamic equations in generalized curvilinear coordinates. In each equation, the coefficients before the two detivatives of surface elevations have different orders of magnitude, i. e., the derivative with the larger ceefficient rnay play a more important role than that with the smaller one. With this advantage, the ADI scheme can then be easily employed to improve the numerical stability and decrease the calculating quantity. The calculation in a harbour and a channel in Macau nearshore area shows that the implicit model is effective in calculating current fields in small size areas.

关键词： Numerical model contravariant component of velocity vector implicit scheme

TWO-DIMENSIONAL PLANE WATER FLOW AND WATER QUALITY DISTRIBUTION IN BOSTEN LAKE

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Journal of Hydrodynamics 2003年第5期15卷 91-97页

作者： FengMin-quan ZhouXiao-de ZhengBang-min MinTao ZhaoKe-yu InstiuteofWaterResourcesandHydroelectricEngineering Xi‘anUniversityofTechnologyXi’an710048China SchoolofwaterResourceandHydropower WuhanUniversityWuhan430072China

The two-dimensional plane water flow and water quality was developed by usingthe techniques of coordinate transformation, alternating directions, staggered grid, linearrecurrence, and implicit scheme in the study of large water body in lakes. The model was proved tobe suitable for treating the irregular boundary and predicting quickly water flow and water *** application of the model to the Bosten Lake in Xinjiang Uygur Autonomous Region of China showsthat it is reasonable and practicable.

关键词： circulation water quality coordinate transformation alterating direction staggered grid implicit scheme

Efficient Finite Difference Methods for the Numerical Analysis of One-Dimensional Heat Equation

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Journal of Applied Mathematics and Physics 2023年第10期11卷 3099-3123页

作者： Md. Shahadat Hossain Mojumder Md. Nazmul Haque Md. Joni Alam Department of Mathematics Comilla University Kotbari Bangladesh Department of Mathematical Sciences Kent State University Kent OH USA

In this paper, we investigate and analyze one-dimensional heat equation with appropriate initial and boundary condition using finite difference method. Finite difference method is a well-known numerical technique for obtaining the approximate solutions of an initial boundary value problem. We develop Forward Time Centered Space (FTCS) and Crank-Nicolson (CN) finite difference schemes for one-dimensional heat equation using the Taylor series. Later, we use these schemes to solve our governing equation. The stability criterion is discussed, and the stability conditions for both schemes are verified. We exhibit the results and then compare the results between the exact and approximate solutions. Finally, we estimate error between the exact and approximate solutions for a specific numerical problem to present the convergence of the numerical schemes, and demonstrate the resulting error in graphical representation.

关键词： Explicit scheme implicit scheme C-N scheme CFL Condition