摘要
在自然科学的许多领域中,如热传导以及其它扩散现象、某些生物形态、化学反应等等,是用抛物型方程或方程组描述的.在一些需要快速计算的大型复杂的科学工程计算问题中,需要利用它们内部的并行性,设计出合理的并行算法,然后在并行机上用并行算法求解.通常这些方程必须通过有限差分法求解.因此,我们需要不断完善和改进已有的传统的差分方法,针对具体问题,构造出合理的具有并行性的新的差分算法.
本文以简单的一维热传导方程为例,利用Saul’yev非对称格式,其中, r =ΔΔxt2,构造出分组显示GE法, GE法在r =ΔΔxt2≤1时稳定,其截断误差比单独使用非对称格式有了明显改善,误差为O (Δt +Δx).在不同的时间层交替使用GEL和GER法,得到交替分组显式AGE方法,其数学描述如下其中, k =0,2,4,L.通过研究得到AGE方法是绝对稳定的,稳定性得到了基本改善,同时其截断误差分别为交替分组显式AGE方法的截断误差为O (Δt +Δx).
进而,将AGE方法推广到求解二维抛物型方程有限差分的并行计算中,该方法具有并行性且是无条件稳定的.以二维扩散抛物型方程的初边值问题为例:边界条件为初始条件为u ( x,y,0)= f(x,y) 0< x,y<1 AGE方法定义为:其中, k =1,2,L.
交替差分块方法是受到解二维问题AGE差分方法的启发而产生的一类新的解决二维问题的方法,在一定条件下这种方法能进行并行计算,并且稳定性较好.
最后,针对一维抛物方程,依据分组交替的思想用Saul’yev非对称格式建立了一种新的具有并行性质的差分算法,新算法的一般描述为:其中增长矩阵矩阵G1和G 2为非负定的,利用Kellogg引理得出新算法是绝对稳定的.
最后,通过数值实验对AGE算法和新算法进行了比较,当r <1, =1或> 1时,可以看出新算法都是收敛的,其误差都是可控的.这从实验的角度证明了前面对新算法稳定性的理论分析,即新算法是绝对稳定的.
In many areas of natural sciences, such as heat conduction, other diffusion phenomena, certain biological morphology and chemical reactions, they are described through parabolic equation or equations. In some large complex scientific and engineering computing problems, they need fast computation. Reasonable parallel algorithm should be designed according to their internal parallelism, and then the answer will be obtained with parallel algorithm in parallel machine. Generally, these equations must be solved by the finite difference method. Therefore, those existed and traditional difference methods have to be constantly refined and perfected. New difference algorithm with parallelism should be constructed reasonably for specific questions.
A simple one-dimensional parabolic equation is used as an example, Using Saul’yev Asymmetric format Among them, r =ΔΔxt2, GE grouped method is constructed, When r =ΔΔxt2≤1, GE is stable. Compared with the non-symmetric form alone, the truncation error has significantly improved, and the error is o (Δx+Δt).
When GEL and GER are used alternatively at different time layers, AGE will be obtained. The description of Alternating Group Explicit is as follows:
The research shows that AGE method is absolutely stable. The truncation errors are showed respectively in the following expressions:
The truncation error of AGE is o (Δx+Δt).
Then, the AGE method is extended to solve the two-dimensional finite difference parabolic equations in parallel computing, the method contains parallelism and is unconditionally stable. Take the initial boundary value problem of two-dimensional parabolic diffusion as an example: Boundary conditions are Initial condition is u ( x,y,0)= f(x,y) 0< x,y<1 AGE method can be defined as
Alternating difference block methods is a new method used to solve two-dimensional problem inspired by the solution method of AGE difference. It makes use of the difference relationship between adjacent grid points to solve the problem. It includes the differential block Law and complementary difference block methods. Under certain conditions, this method is capable of doing parallel computing, and has good stability.
For one-dimensional parabolic equation, according to the ideological group alternate format by Saul’yev asymmetric, a new difference algorithm with parallelism is established, which is generally described as: where Growth Matrix:
Matrix G1 and G 2 set are non-negative. The new algorithm is absolutely stable according to Kellogg lemma.
Finally, numerical experiments are done to compare the AGE algorithm and new algorithm, when r <1, =1 or > 1, the new algorithms are convergent and its errors are controllable. The previous theoretical analysis on the stability of the new algorithm is proved, namely, the new algorithm is absolutely stable.
引文
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