Higher order ADI method with completed Richardson extrapolation for solving unsteady convection–diffusion equations

Dai, Ruxin; Zhang, Jun; Wang, Yin

doi:10.1016/j.camwa.2015.12.007

Cited by 10 publications

(5 citation statements)

References 23 publications

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“…Table 2 indicates that the IELDTM provides much more accurate results with far less dof than the ADI schemes. In Figures 4 and 5, spectral convergence of the IELDTM has been proven with respect to K − refinement (order refinement) and Δx=Δy − refinement for various values of the direction parameters θ x and θ y : As theoretically expected, K − refinement procedure leads to an exponential convergence and Δx=Δy − refinement approach leads to a polynomial convergence like Δh P when Δx ¼ Δy ¼ Δh: Since K − refinement procedure increases the local degrees of freedom (ldof) linearly as ldof (Dai et al, 2016) at t f ¼ 0:25, N ¼ 15; S ¼ 14 and K ¼ 8 for Problem 2 EC 40,9/10 IELDTM provides highly accurate results with optimized dof. The mesh discretization parameter S plays a vital role in the accuracy and computational cost of the IELDTM.…”

Section: Numerical Experimentsmentioning

confidence: 59%

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A spatial local method for solving 2D and 3D advection-diffusion equations

Tunc,

Sari

2023

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PurposeThis study aims to derive a novel spatial numerical method based on multidimensional local Taylor series representations for solving high-order advection-diffusion (AD) equations.Design/methodology/approachThe parabolic AD equations are reduced to the nonhomogeneous elliptic system of partial differential equations by utilizing the Chebyshev spectral collocation method (ChSCM) in the temporal variable. The implicit-explicit local differential transform method (IELDTM) is constructed over two- and three-dimensional meshes using continuity equations of the neighbor representations with either explicit or implicit forms in related directions. The IELDTM yields an overdetermined or underdetermined system of algebraic equations solved in the least square sense.FindingsThe IELDTM has proven to have excellent convergence properties by experimentally illustrating both h-refinement and p-refinement outcomes. A distinctive feature of the IELDTM over the existing numerical techniques is optimizing the local spatial degrees of freedom. It has been proven that the IELDTM provides more accurate results with far fewer degrees of freedom than the finite difference, finite element and spectral methods.Originality/valueThis study shows the derivation, applicability and performance of the IELDTM for solving 2D and 3D advection-diffusion equations. It has been demonstrated that the IELDTM can be a competitive numerical method for addressing high-space dimensional-parabolic partial differential equations (PDEs) arising in various fields of science and engineering. The novel ChSCM-IELDTM hybridization has been proven to have distinct advantages, such as continuous utilization of time integration and optimized formulation of spatial approximations. Furthermore, the novel ChSCM-IELDTM hybridization can be adapted to address various other types of PDEs by modifying the theoretical derivation accordingly.

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Section: Numerical Experimentsmentioning

confidence: 59%

“…We compare the central, backward and forward IELDTM with the Richardson extrapolated alternating implicit direction (ADI-CRE) method proposed by Dai et al (2016) in Table 2. Table 2 indicates that the IELDTM provides much more accurate results with far less dof than the ADI schemes.…”

Section: Numerical Experimentsmentioning

confidence: 99%

A spatial local method for solving 2D and 3D advection-diffusion equations

Tunc,

Sari

2023

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“…Sun, Zhang and Dai developed a sixth-order FD scheme for solving the 2D convectiondiffusion equation [37,38]. In their approach, the ADI method is applied to compute the fourth-order accurate solution on the fine and coarse grids, respectively, then the Richardson extrapolation technique and an operator-based interpolation scheme are employed in each ADI iteration to calculate the sixth-order accurate solution on the fine grid.…”

Section: Another Group Of Researchers Made Efforts To Design An Iteramentioning

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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“…Therefore, every effort has been put to develop the efficient and stable numerical techniques. To solve the convection-diffusion equation, a huge body of research on proficient numerical methods, such as the finite element method (see, e.g., [3][4][5]), the finite difference method (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]), and others, has been created. We now concentrate on the research about the finite difference method.…”

Section: Introductionmentioning

confidence: 99%

“…On this account, its great effectiveness on solution accuracy and computational efficiency was demonstrated. All of those methods are second-order accurate in time and fourth-order accurate in space (see more [18][19][20]). Additional developments on a series of ADI methods have recently been completed.…”

Section: Introductionmentioning

confidence: 99%

A mass-conservative higher-order ADI method for solving unsteady convection–diffusion equations

2020

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In the paper, a high-order alternating direction implicit (ADI) algorithm is presented to solve problems of unsteady convection and diffusion. The method is fourth- and second-order accurate in space and time, respectively. The resulting matrix at each ADI computation can be obtained by repeatedly solving a penta-diagonal system which produces a computationally cost-effective solver. We prove that the proposed scheme is mass-conserved and unconditionally stable by means of discrete Fourier analysis. Numerical experiments are performed to validate the mass conservation and illustrate that the proposed scheme is accurate and reliable for convection-dominated problems.

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Higher order ADI method with completed Richardson extrapolation for solving unsteady convection–diffusion equations

Cited by 10 publications

References 23 publications

A spatial local method for solving 2D and 3D advection-diffusion equations

A spatial local method for solving 2D and 3D advection-diffusion equations

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

A mass-conservative higher-order ADI method for solving unsteady convection–diffusion equations

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