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

Wang, Qinghe; Pan, Kejia; Hu, Hongling

doi:10.1186/s13662-018-1591-1

Cited by 3 publications

(2 citation statements)

References 40 publications

(57 reference statements)

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“…We need point out that the fourth-order compact difference method, which is employed to discretize the differential equation, can be replaced by other numerical methods, such as the FE method and the finite volume method. Furthermore, we also plan to extend this study and apply EXFMG algorithm with sixth-order combined compact difference scheme [43][44][45] to solve unsteady three-dimensional convection-diffusion equations.…”

Section: Resultsmentioning

confidence: 99%

An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection–diffusion equations

Zheng

Pan

2018

Adv Differ Equ

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In this paper, we propose an extrapolation full multigrid (EXFMG) algorithm to solve the large linear system arising from a fourth-order compact difference discretization of two-dimensional (2D) convection diffusion equations. A bi-quartic Lagrange interpolation for the solution on previous coarser grid is used to construct a good initial guess on the next finer grid for V-or W-cycles multigrid solver, which greatly reduces the number of relaxation sweeps. Instead of performing a fixed number of multigrid cycles as used in classical full multigrid methods, a series of grid level dependent relative residual tolerances is introduced to control the number of the multigrid cycles. Once the fourth-order accurate numerical solutions are obtained, a simple method based on the midpoint extrapolation is employed for the fourth-order difference solutions on two-level grids to construct a sixth-order accurate solution on the entire fine grid cheaply and directly. Numerical experiments are conducted to verify that the proposed method has much better efficiency compared to classical multigrid methods. The proposed EXFMG method can also be extended to solve other kinds of partial differential equations.

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Section: Resultsmentioning

confidence: 99%

An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection–diffusion equations

Zheng

Pan

2018

Adv Differ Equ

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“…Moreover, the unique solvability of the CCD system is obtained under non-periodic boundary conditions. Existing theoretical works on the CCD method are focused on solving the differential equations subject to periodic boundary conditions [36,46]. The convergence analysis for the CCD scheme will be our future objective.…”

Section: Discussionmentioning

confidence: 99%

A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

Pan

Yue

et al. 2020

Comp. Appl. Math.

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In this paper, an efficient and high-order accuracy finite difference method is proposed for solving multidimensional nonlinear Burgers' equation. The third-order three stage Runge-Kutta total variation diminishing (TVD) scheme is employed for the time discretization, and the three-point combined compact difference (CCD) scheme is used for spatial discretization. Our method is third-order accurate in time and sixth-order accurate in space. The CCD-TVD method treats the nonlinear term explicitly thus it is very efficient and easy to implement. In addition, we prove the unique solvability of the CCD system under non-periodic boundary conditions. Numerical experiments including both two-dimensional and three-dimensional problems have been conducted to demonstrate the high efficiency and accuracy of the proposed method.

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A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

Chen

Pan

2018

International Journal of Computer Mathematics

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

Cited by 3 publications

References 40 publications

An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection–diffusion equations

An extrapolation full multigrid algorithm combined with fourth-order compact scheme for convection–diffusion equations

A spatial sixth-order CCD-TVD method for solving multidimensional coupled Burgers’ equation

A CCD-ADI method for two-dimensional linear and nonlinear hyperbolic telegraph equations with variable coefficients

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