一维双曲型方程的组合差商法及其在二维中的推广

【作者】 刘轶中；

【导师】 张大凯；

【作者基本信息】 贵州大学 ， 基础数学， 2007， 硕士

【摘要】 本文针对一维和二维双曲型方程的初边值问题，设计了几类高效率串行格式和并行算法。首先，运用组合差商算法给出了求解一维双曲型方程的一类显式差分格式，其精度一般为o(τ~3+h~3)，最高精度为o(τ~4+h~4)。其次，构造了求解双曲型方程u_t+au_x=0的初边值问题的一组含双参数的分组并行算法(GE、GEL、GER)，格式的局部截断误差阶一般为o(τ+h)，当β=1、1-(4/r~2)＜α＜1时，稳定性条件为r＞0；当β=1、1-(4/r~2)=α＜1时，稳定性条件为r＞0且r≠1。特别当α=1/2、β=r-1/2r时，GE、GEL、GER格式的局部截断误差阶为o(τ~2+h~2)，稳定性条件为0＜r≤4/3。最后，构造了求解二维双曲型方程u_t+au_x+bu_y=0的初边值问题的一组分组并行算法(GE、GEL、GER)，格式的局部截断误差阶一般为o(τ+h)，稳定性条件为0＜r≤1。本文对各格式都进行了数值例子计算，验证了理论分析的结果。本文所构造的差分格式较以往的格式，精度有了很大的提高，稳定性条件也比较好。更多还原

【Abstract】 In this paper, I design a series of high-efficiency serial schemes and parallel algorithms for one-dimension and two-dimension hyperbolic equation.Firstly, a new kind of explicit difference scheme for one-dimension hyperbolic equation is proposed. The truncation error of the scheme is of ordero(τ~4 + h~4) .Secondly, a class of group parallel algorithms(GE、GEL、GER) containing biparameters are constructed for solving the hyperbolio equation u_t + au_x = 0. Thelocal truncation error is always of order o(τ+ h) ,The stability condition is r > 0withβ=1、1- 4/r~2<α<1 The stability condition is r>0 and r≠1 withβ=1、1-4/r~2 =α< 1. The local truncation error is of order o(τ~2 + h~2) withα= 1/2、β=(r-1)/(2r), The stability condition is 0 < r≤4/3 withα=1/2、β= (r-1)/(2r). In the end,a class of group parallel algorithms(GE、GEL、GER) are constructed for solving the two-dimensional hyperbolio equation u_t + au_x +bu_y=0 in this paper. The localtruncation error is always of order o(τ+ h) ,The stability condition is 0 < r≤1. Inthis paper, every type scheme is given numerical compute, and the result validates the theory analysis.In this paper, the difference schemes are better than those of before at precise and stability.更多还原