Unconditional stability of alternating difference schemes with intrinsic parallelism for two-dimensional parabolic systems

Yuan, Guangwei; Shen, Longjun; Zhou, Youhe

doi:10.1002/(sici)1098-2426(199911)15:6<625::aid-num2>3.0.co;2-o

Cited by 19 publications

(17 citation statements)

References 8 publications

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“…An alternative approach of parallelization is to take advantage of previous time steps since the problem considered here is time dependent; see [3,7,12,13,14] for related discussions. Kuznetsov [7] proposed a modified approximation scheme of mixed type, where the standard second order implicit scheme is used inside each subdomain, while the explicit Euler scheme is applied to obtain the interface values on the new time level.…”

Section: Ams Subject Classifications 65n10 65p05mentioning

confidence: 99%

Efficient Parallel Algorithms for Parabolic Problems

Du¹,

Mu²,

Wu³

2002

SIAM J. Numer. Anal.

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Abstract. Domain decomposition algorithms for parallel numerical solution of parabolic equations are studied for steady state or slow unsteady computation. Implicit schemes are used in order to march with large time steps. Parallelization is realized by approximating interface values using explicit computation. Various techniques are examined, including a multistep second order explicit scheme and a one-step high-order scheme. We show that the resulting schemes are of second order global accuracy in space, and stable in the sense of Osher or in L∞. They are optimized with respect to the parallel efficiency.Key words. parabolic equations, finite difference, parallel efficiency, stability, approximation accuracy AMS subject classifications. 65N10, 65P05PII. S00361429003817101. Introduction. Domain decomposition is a powerful tool for devising parallel PDE methods. There is rich literature on domain decomposition methods for both elliptic and time-dependent problems [2,6]. We consider the linear parabolic problem in this paper. Explicit schemes are often naturally parallel and also easy to implement, but they usually require small time steps because of stability constraints. Implicit schemes are necessary for finding steady state solutions or computing slowly unsteady problems where one needs to march with large time steps; however, implicit schemes are not inherently parallel because at each time step essentially an elliptic type of problem needs to be solved.A conventional approach of parallelizing the implicit schemes is to apply the elliptic-type domain decomposition based preconditioning methods to the problem arising from the semidiscretization at each time step. It is noted [1] that the resulting problem is well conditioned when the time step is small; nevertheless, small step size is not always desirable in situations where implicit schemes become necessary to use.An alternative approach of parallelization is to take advantage of previous time steps since the problem considered here is time dependent; see [3,7,12,13,14] for related discussions. Kuznetsov [7] proposed a modified approximation scheme of mixed type, where the standard second order implicit scheme is used inside each subdomain, while the explicit Euler scheme is applied to obtain the interface values on the new time level. Once the interface values are available, the global problem is fully decoupled and can thus be computed in parallel. A similar hybrid scheme was proposed in [3], where instead of using the same spacing h as for the interior points where the implicit scheme is applied, a larger spacing H D is used at each interface point where the explicit scheme is applied. Due to stability and accuracy requirements, both methods do not lead to satisfactory parallel efficiency as shown

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Section: Ams Subject Classifications 65n10 65p05mentioning

confidence: 99%

Efficient Parallel Algorithms for Parabolic Problems

Du¹,

Mu²,

Wu³

2002

SIAM J. Numer. Anal.

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show abstract

“…Many powerful methods have been presented in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]20]. In 1983, Evans and Abdullah [1] observed that the alternate use of different schemes with truncation errors of opposite signs can implement the parallel computation and has high accuracy and unconditional stability.…”

Section: Introductionmentioning

confidence: 99%

“…After that, the alternating technology became very popular method for a large number of equations, such as the AGE, ASE-I, ASC-N schemes for solving dispersive equation [4][5][6][7][8]10], KdV equation [9], forth-order parabolic equation [11,12], and so on. In particular, Yuan et al [13,14] first constructed the general schemes with intrinsic parallelism and variable time steplengthes for one-dimensional and two-dimensional parabolic systems, which cover some well-known schemes. Unconditional stability and numerical experiments were given, and numerical results show that the schemes with variable time steplengthes are in preference to the scheme with equal time steplengthes.…”

Section: Introductionmentioning

confidence: 99%

Unconditional stability of alternating difference schemes with variable time steplengthes for dispersive equation

Guo

Lü

Liu

2015

Applied Mathematics and Computation

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“…Many effective methods have been presented in [1][2][3][4][5][6][7][8][9][10][11]. In 1983, Evans and Abdullah [1] observed that the alternate use of different schemes with truncation errors of opposite signs can lead to the cancelation of error terms at most points on the mesh lines, they first developed the Alternating Group Explicit (AGE) scheme for solving the parabolic equation.…”

Section: Introductionmentioning

confidence: 99%

“…Zhou [4] and Zhou et al [5] advocated the concept ''intrinsic parallelism'' for the first time, we call a difference scheme that has a direct and natural algorithm of parallel character a difference scheme with ''intrinsic parallelism''. In 1999, Yuan et al [6] constructed the general schemes with intrinsic parallelism for two-dimensional parabolic systems, and gave the unconditionally stable analysis. More recently, the AGE scheme, the ASE-I scheme and the ASC-N scheme were extended to the dispersive equation and the nonlinear KdV equation in [7][8][9][10][11], and numerical results are satisfied.…”

Section: Introductionmentioning

confidence: 99%