A rational high-order compact ADI method for unsteady convection–diffusion equations

Tian, Zhenfu

doi:10.1016/j.cpc.2010.11.013

Cited by 34 publications

(26 citation statements)

References 26 publications

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“…Equation 2, which is often regarded as the linearized version of 1D Navier-Stokes equation, describes convection and diffusion of various physical properties such as mass, heat, energy, and vorticity. It is encountered in many fields of science and engineering [33,34]. Therefore, it is of great importance to develop accurate and stable numerical methods for solving the convection-diffusion equations.…”

Section: Remark 1 Consider the Unsteady 1d Convection-diffusion Equationmentioning

confidence: 99%

Unique solvability of the CCD scheme for convection–diffusion equations with variable convection coefficients

Wang

Pan

2018

Adv Differ Equ

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The combined compact difference (CCD) scheme has better spectral resolution than many other existing compact or noncompact high-order schemes, and is widely used to solve many differential equations. However, due to its implicit nature, very little theoretical results on the CCD method are known. In this paper, we provide a rigorous theoretical proof for the unique solvability of the CCD scheme for solving the convection-diffusion equation with variable convection coefficients subject to periodic boundary conditions.

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Section: Remark 1 Consider the Unsteady 1d Convection-diffusion Equationmentioning

confidence: 99%

Unique solvability of the CCD scheme for convection–diffusion equations with variable convection coefficients

Wang

Pan

2018

Adv Differ Equ

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“…to transform the physical (x, y) plane into a computational (ξ, η) plane. Under this transformation equations (24) and (25) become…”

Section: Extension To the 2d Incompressible Navier-stokes Equationsmentioning

confidence: 99%

Fourth order compact schemes for variable coefficient parabolic problems with mixed derivatives

Sen

2016

Computers & Fluids

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In this article, we have developed a higher order compact numerical method for variable coefficient parabolic problems with mixed derivatives. The finite difference scheme, presented here for two-dimensional domains, is based on fourth order spatial discretization. The time discretization has been carried out using using second order Crank-Nicolson. The present scheme shows good dispersion relation preserving property and has been thoroughly investigated for stability. The discrete Fourier analysis shows that the method is unconditionally stable. The fact that the method has been particularly developed for parabolic equations with mixed derivatives makes it suitable for solving incompressible Navier-Stokes (N-S) equations in irregular domains. To verify the proposed method, several problems with exact and benchmark solutions has been investigated. The proposed compact discretization has been extended to tackle flows of varying complexities governed by the twodimensional unsteady N-S equations in domain beyond rectangular. The results show good agreement for all the problems considered.

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“…However, it is still expensive to be used at each time step, especially for higher dimensional problems. The other strategy is to develop efficient and cost-effective difference algorithms, such as alternating direction implicit (ADI) [11,15,16,18,[20][21][22][23][25][26][27][28] or locally one dimensional (LOD) procedures [29][30][31][32], which are based on reducing high dimensional problems in several space variables to collections of 1D problems and only requiring to solve tridiagonal matrices, are simple to implement and economical to use. It is well known that the classical ADI method developed by Peaceman and Rachford [15], and Douglas and Gunn [16] have been popular due to their computational cost-effectiveness.…”

Section: Introductionmentioning

confidence: 99%

“…It is well known that the classical ADI method developed by Peaceman and Rachford [15], and Douglas and Gunn [16] have been popular due to their computational cost-effectiveness. However, these two ADI schemes, which are second order accurate in space and often produce significant dissipation and phase error [11,21,22], are not ideally suited to deal with the spatial discretization of convection-dominated transport problems. And the Peaceman-Rachford ADI scheme for 3D convection diffusion problems is conditionally stable.…”

Section: Introductionmentioning

confidence: 99%

“…To achieve higher spatial accuracy computational cost-effectiveness, recently, there has been a renewed interest in the development of HOC ADI methods for 2D and 3D unsteady convection diffusion equations. Karra and Zhang [20], You [21], Tian and Ge [22], Tian [11], developed, respectively, an HOC ADI method for solving 2D unsteady convection diffusion equations with constant coefficients. They are all second order accurate in time, fourth order accurate in space and unconditionally stable.…”

Section: Introductionmentioning

confidence: 99%

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A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems

Ge¹

2018

ACM

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In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa's HOC ADI method [26], Cao and Ge's HOC ADI method [27], and our previous exponential HOC ADI method [28].

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A rational high-order compact ADI method for unsteady convection–diffusion equations

Cited by 34 publications

References 26 publications

Unique solvability of the CCD scheme for convection–diffusion equations with variable convection coefficients

Unique solvability of the CCD scheme for convection–diffusion equations with variable convection coefficients

Fourth order compact schemes for variable coefficient parabolic problems with mixed derivatives

A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems

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