抛物型方程的高精度高稳定性格式及其并行算法的研究

发布时间：2018-04-10 00:00

本文选题：抛物型方程　切入点：组合差商法　出处：《贵州大学》2007年硕士论文

【摘要】： 本文针对抛物型方程的初边值问题，采用组合差商法和参数的应用。设计了几类高精确度串行格式和并行算法。也为算法的构造提供了灵活有效的构造方法和新的思路。对于串行算法：我们在空间节点宽度为3，时间层宽度为3的局部节点集上构造了一个高精度的三层七点显格式，其精度达到O(τ~4+h~6)，稳定性条件是r≤1/6。当网比r取特定值时，格式的精度可提高到O(τ~4+h~8)。我们还在空间节点宽度为3，，时间层宽度为3的局部节点集上构造了一类高精度的三层九点含参数隐式差分格式，其截断误差达到O(τ~3+h~6)，绝对稳定。当参数取特定值时，格式的截断误差可以提高到O(τ~4+h~8)。稳定性条件是0＜r＜(1/(20(1/2)))。当r取特定值时，格式的截断误差可达到O(τ~5+h~(10))。对于并行格式：我们用组合差商法构造了带一个参数的两层六点半显格式和它的对称格式。然后利用这两个格式建立求解抛物型方程的一类在空间方向并行、时间方向步进的带参数的分组显式GE并行算法，该算法的截断误差为O(τ+h~2)，条件稳定。当参数取特定值时，该算法的截断误差可提高到O(τ~2+h~3)。当参数取零、网比r取特定值时，该算法的截断误差可达到O(τ~2+h~4)。当空间节点为奇数时。构造了GEL格式和GER格式。我们还构造了一类新型的不仅在空间上可以并行，在时间上也可以并进的时空并行格式，该算法很大地提高了算法并行度，发展了传统的只能在空间并行而在时间上是步进的并行算法。提出了研究算法的一些新思路。
[Abstract]:Based on the parabolic equation with initial boundary value problem, using combined difference quotient and parameter design. Several kinds of high accuracy serial format and parallel algorithm. It provides a flexible and effective construction method and new ideas to construct the algorithm.
For the serial algorithm:
We in the space node width is 3, time width of 3 local node set on the structure of a high precision three layer seven point explicit scheme, the accuracy of O (~4+h~6), the stability condition is r less than 1/6. when the net ratio R specific value, the accuracy can be improved to O format (tau ~4+h~8).
We are the space node width is 3, time width of 3 local node sets to construct a class of high precision three layer nine parametric implicit difference scheme, the truncation error is O (~3+h~6), absolute stability. When the parameters of the specific value, the truncation error format can be improved O (~4+h~8). The stability condition is 0 (r < < 1/ (20 (1/2))). When the R specific value, the truncation error format can reach O (~5+h~ (10)).
For the parallel format:
We use a combination of difference quotient is two layers structure and its symmetry half past six explicit format with one parameter. Then by using a class set for solving the parabolic equation of the two schemes in parallel in space, time step packet parameters with explicit GE parallel algorithm, the truncation error of the algorithm is O (+h~2), conditional stability. When the parameters of the specific value, the truncation error of this algorithm can be improved to O (~2+h~3). When the parameter is zero, the net ratio R specific value, the truncation error of this algorithm can reach O (~2+h~4). When the node space is odd constructed. GEL format and GER format.
We also construct a new kind of not only in space can be parallel, in time and space can also be into parallel format, this algorithm greatly improves the algorithm parallelism, the development of the traditional space only in parallel and in time is stepping the parallel algorithm. This paper puts forward some new ideas and research method.

【学位授予单位】：贵州大学
【学位级别】：硕士
【学位授予年份】：2007
【分类号】：O241.82

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