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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

对流占优扩散问题的一个新的并行交替方向

算法

冯慧，薛冠宇

武汉大学数学与统计学院，武汉，430072

摘要：本文针对对流占优扩散问题提出了一个并行的交替方向算法，算法的第一个计算方向包

括两个区域分解算法，当第 n 时间层的数值解已知时，分别利用两算法计算得到两个数值解，

再取其平均值作为第 (n +

) 时间层的数值解；同理，算法的另一个计算方向根据所求得的第

(n +

) 时间层的数值解来计算得到第 (n + 1) 时间层数值解. 新算法在保持了并行性和无条件

稳定性的同时，又得到了较高的计算精度. 误差分析和数值算例揭示了新算法的精度和有效性.

关键词：对流占优扩散问题, 交替方向, 有限差分, 无条件稳定性, 并行性

中图分类号： O241.8

A new parallel alternating-direction algorithm

for convection-dominated diﬀusion problems

Hui Feng , Guanyu Xue

School of Mathematics and Statistics, Wuhan University, Wuhan 430072

Abstract: In this paper, a new parallel alternating-direction algorithm for

convection-dominated diﬀusion problems is proposed. One direction calculation of the new

algorithm includes two domain decomposition methods that are applied to compute the

values at (n +

)st time level by use of known numerical solutions at nth time level,

respectively, then the average of two above values is chosen to be the numerical solutions at

(n +

)st time level. Next, the other direction calculation of the new algorithm will be used to

solve the numerical solutions at (n + 1)st time level by use of known values at (n +

)st time

level above by the same technique. The new algorithm obtains high accuracy while

maintaining parallelism and unconditional stability. Both error analysis and numerical

examples show the accuracy and eﬃciency of the new algorithm.

Key words: convection-dominated diﬀusion problems, alternating-direction, ﬁnite diﬀerence,

unconditional stability, parallelism

Foundations: The Doctoral Fund of Ministry of Education of China(20130141110026)，the National Natural Science

Foundation of China (91130022)

Author Introduction: Hui Feng （1968-）， male， professor，major research direction： computational mathematics.

Guanyu Xue（1988-），male，doctoral student，major research direction：computational mathematics.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

0 Introduction

In this paper, we consider the convection-dominated diﬀusion problems in two dimensions

∂u

∂t

− ∇ · (a∇u) + ∇ · bu = f(X, t), X ∈ Ω, t ∈ (0, T ], (1)

u(X, 0) = u

(X), X ∈ Ω, (2)

u(X, t) = 0, X ∈ ∂Ω, t ∈ (0, T ], (3)

where |b| ≫ a, a > 0 is a constant, b is the convection coeﬃcient and the domain Ω ∈

, x

) × (y

, y

) with the boundary is ∂Ω.

Convection-diﬀusion processes appear in many areas of science and technology, such as ﬂuid

dynamics, hydrology, the phenomena of heat and mass transfer and so on. In many convection-

diﬀusion types of problems arising from real applications, the convection coeﬃcients are often

much larger than the diﬀusion ones, leading to convection-dominated problems. The numerical

method for such problems presents a challenging computational task. Many standard methods,

such as the standard ﬁnite diﬀerence method or ﬁnite element method perform poorly and

exhibit severe nonphysical oscillations for convection-dominated problems. The investigation

of convection-dominated diﬀusion problems has been an interesting object recently [1, 2, 3, 4,

5, 6, 7, 8]. However, parallel algorithm for convection-dominated diﬀusion problems is few.

Parallel algorithm is a new approach for high performance computing. Recently, the

parallel diﬀerence method for parabolic equations has been studied rapidly. Evans [9] ﬁrst

constructed the Alternating Group Explicit (AGE) scheme for the diﬀusion equation. The

Alternating Segment Explicit-Implicit (ASE-I) scheme and the Alternating Segment Crank-

Nicolson (ASC-N) scheme were designed in [10, 11]. Afterwards, the alternating segment algo-

rithms (AGE scheme, ASE-I scheme and ASC-N scheme) above became very eﬀective methods

for parabolic equations, such as convection-diﬀusion equation [12, 13, 14], dispersive equation

[15, 16, 17, 18, 19], forth-order parabolic equation [20, 21, 22] and nonlinear third-order KdV

equation [23]. Meanwhile, domain decomposition methods (DDMs) for partial diﬀerential e-

quations have been studied extensively [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. In 1999, the

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

were presented in [35], the unconditional stability analysis was given. The unconditionally

stable domain decomposition method was obtained by the alternating technique in [36, 37, 38].

In fact, alternating segment algorithm is also a form of domain decomposition method, which

is not only suitable for parallel computation but also unconditionally stable.

Inspired by alternating segment algorithm, a new parallel algorithm for convection-dominated

diﬀusion problems will be presented in this paper by alternating direction implicit (ADI) tech-

- 2 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

nique [39]. The new parallel algorithm along x-direction calculation and y-direction calculation

consist of two DDMs, respectively. The new algorithm can be stated as follows:

1. along x-direction calculation

(1). DDM I is applied to compute the values at (n +

)st time level noted as V

by use

of known value U

at nth time level.

(2). DDM II is applied to compute the values at (n +

)st time level noted as W

by use

of known value U

at nth time level.

(3). In order to improve accuracy, let U

= (V

+ W

)/2 be the numerical solutions

at (n +

)st time level.

2. along y-direction calculation

(1). DDM I is applied to compute the values at (n + 1)st time level noted as V

n+1

by use

of known value U

at (n +

)st time level.

(2). DDM II is also applied to compute the values at (n + 1)st time level noted as W

n+1

use of known value U

at (n +

)st time level.

(3). In order to improve accuracy, let U

n+1

= (V

n+1

+ W

n+1

)/2 be the numerical solutions

at (n + 1)st time level.

This paper is arranged as follows: In Section 2, we introduce a modiﬁed Crank-Nicolson

scheme and four Saul’yev asymmetric diﬀerence schemes to construct two DDMs, then the

detailed parallel algorithm presentations are given. The error analysis and unconditional stabil-

ity of the new algorithm are discussed in Sections 3. Finally, we give some numerical examples,

which illustrate the accuracy and eﬃciency of the new algorithm proposed in this paper.

1 Algorithm presentation

Let τ = T /N be the temporal step sizes and t

= nτ for n = 0, 1, . . . , N , where N is

positive integer. Denote ∆x and ∆y as the spatial step sizes in x and y directions, respectively,

where ∆x = (x

− x

)/N

and ∆y = (y

− y

)/N

, N

and N

are positive integers. The

grid points (x

, y

) in space are deﬁned as x

= x

+ i∆x (i = 0, 1, . . . , N

) and y

= y

+ j∆y

(j = 0, 1, . . . , N

), respectively. Let U

i,j

approximate u

i,j

for n > 0 at (x

, y

, t

). For simplicity,

we introduce the diﬀerence operators will be used in this paper as follows:

i,j

i+1,j

− U

i,j

∆x

, δ

¯x

i,j

− U

i−1,j

∆x

, δ

ˆx

i,j

i+1,j

− U

i−1,j

2∆x

i,j

i,j+1

− U

i,j

∆y

, δ

¯y

i,j

− U

i,j−1

∆y

, δ

ˆy

i,j

i,j+1

− U

i,j−1

2∆y

i,j

n+1

i,j

− U

i,j

, δ

ˆx

i,j

n+1

i+1,j

− U

i−1,j

2∆x

, δ

¯x

i,j

i+1,j

− U

n+1

i−1,j

2∆x

Therefore, based on the notations above, we can propose the new parallel alternating-

direction algorithm for convection-dominated diﬀusion problems (1)–(3). The process of new

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