Time-Extrapolation Algorithm (TEA) for Linear Parabolic Problem

Hu, Hongling

doi:10.4208/jcm.1310-fe1

Cited by 14 publications

(6 citation statements)

References 18 publications

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“…The iterative method is a popular method used to solve equation systems obtained from discrete methods, such as the finite difference method [1]- [3], the finite volume method [4,5], the finite element method [6]- [9], the lattice Boltzmann method [1,10], and the spectral element method [11,12], due to the ease of code development and low computational time and memory. For solving transient problems, the use of an explicit iterative method has a limitation of time step values.…”

Section: Introductionmentioning

confidence: 99%

A Method of the Lagrange Polynomial Interpolation with Weighting Factors for Reducing Computational Time

Prasopchingchana¹

2021

JMechE

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This paper proposes a novel extrapolation method known as the Lagrange polynomial interpolation with weighting factors (LPI-WF) to determine initial guess values in each time step for solving equation systems of transient problems using iterative methods. The LPI-WF method is developed from the Lagrange polynomial interpolation to determine the direct extrapolation values and multiply them with the weighting factors. The weighting factors are calculated by considering the number of the previous time steps involved in the extrapolation and the duration between the present time step and the previous time steps. Thus, the LPI-WF method is proper for use with highorder temporal schemes. The key advantages of the LPI-WF method are that the computational time required to achieve the steady-state condition of the transient problems is reduced and the computation codes with the LPI-WF method is more stable than without the LPI-WF method at high time step values. A performance test of the LPI-WF method is carried out by comparing the computational time based on the “lid-driven cavity flow” problem for Reynolds numbers of 1000 and 5000. The test result shows that the computational time of the problem when the LPI-WF method is adopted can be reduced up to 10.46 % compared to the conventional method.

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Section: Introductionmentioning

confidence: 99%

A Method of the Lagrange Polynomial Interpolation with Weighting Factors for Reducing Computational Time

Prasopchingchana¹

2021

JMechE

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“…An extrapolation interpolation operator is applied to provide a good initial guess of the iterative solution on the next finer grid, which accelerates the convergence of the original cascadic multigrid algorithm greatly. Hereafter, the EXCMG method has been successfully extended to direct current resistivity modeling problems in geophysical exploration, 21 three‐dimensional (3D) elliptic problems, 22,23 H 2 + α ‐regular (0 < α ≤ 1) elliptic problems, 24 non‐smooth elliptic problems, 25 parabolic problems, 26 and Poisson equations 6,27 . In 2009, Wang and Zhang constructed a multiscale multigrid (MSMG) method for Poisson equations.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth noting that those efficient MGs (EXCMG, MSMG, and EXCMG-MSMG) 4,6,20,21,[23][24][25][26][27][28][29][30] are implemented on uniform grids, since the projection and interpolation operators established on uniform grids. Hence, those MGs are not suitable on nonuniform grids.…”

Section: Introductionmentioning

confidence: 99%

An efficient multiscale‐like multigrid computation for 2D convection‐diffusion equations on nonuniform grids

Zheng

2020

Math Methods in App Sciences

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An efficient multiscale‐like multigrid (MSLMG) method combined with a high‐order compact (HOC) difference scheme on nonuniform grids is presented to solve the two‐dimensional (2D) convection‐diffusion equations. The discrete systems with given appropriate initial solutions on two finest grids are solved to obtain the MSLMG solutions with discretization‐level accuracy by performing fewer multigrid cycles; it is implemented with alternating line Gauss–Seidel smoother, interpolation, and restriction operators on the nonuniform grids. Numerical experiments of boundary layer or local singularity problems are conducted to show that the proposed algorithm with the HOC scheme on nonuniform grids is efficient and effective to decrease the computational cost and time, and the computed approximation on the nonuniform grids has fourth order accuracy.

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“…This method uses a new extrapolation formula to construct a quite good initial guess for the iterative solution on the next finer grid, which greatly improves the convergence rate of the original CMG algorithm (see [24][25][26] for details). Then the EXCMG method has been successfully applied to non-smooth elliptic problems [27,28], parabolic problems [29], and some other related problems [30][31][32]. Moreover, Pan [33] and Li [34,35] developed some EX-CMG methods combined with high-order compact difference schemes to solve Poisson equations.…”

Section: Introductionmentioning

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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Time-Extrapolation Algorithm (TEA) for Linear Parabolic Problem

Cited by 14 publications

References 18 publications

A Method of the Lagrange Polynomial Interpolation with Weighting Factors for Reducing Computational Time

A Method of the Lagrange Polynomial Interpolation with Weighting Factors for Reducing Computational Time

An efficient multiscale‐like multigrid computation for 2D convection‐diffusion equations on nonuniform grids

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

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