【作者】 王小光；

【导师】 刘晓；

【作者基本信息】 河南师范大学 ， 计算数学， 2012， 硕士

【摘要】 高精度紧致差分格式作为数值计算的重要研究问题之一,在很多科学计算领域中占有重要地位.而且,随着工程问题的日趋复杂化,对数值格式的要求越来越高,低阶精度差分格式已经不能完全满足求解问题的要求,因此,有必要构造高分辨率、高精度的差分格式以满足科学和工程计算的需要.本文的研究内容可以分为以下三个方面：第一：依据差分格式的伪波数应该在尽可能大的波数范围内接近物理波数的思想,构造了满足四阶精度且具有高分辨率的三对角四阶紧致差分格式.一方面,它可以与近些年发展的求解(循环)三对角方程组的高效算法相结合,以更高的分辨率、更小的计算量来计算偏微分方程的一阶导数；另一方面,与传统格式相比,该格式的最大精确求解波数可以达到2.5761,大于传统格式的1.13097.因此,优化格式更适合模拟小尺度波动.第二：通过比较我们建立的四阶最优紧致差分格式,以及传统的六阶和八阶紧致差分格式,来研究精度和分辨率之间的关系.一般而言,高精度的差分格式具有较高的分辨率,但是,数值格式精度和分辨率是两个不同的概念,通过差分格式对实际算例模拟的比较,我们发现：针对小尺度波动问题,精度低的四阶格式反而比六阶和八阶的分辨率高.因此,在解决实际问题时,需要选择具有合适的精度和分辨率的数值格式.第三：与传统的四阶紧致差分格式进行对比,说明在实际应用时,我们建立的四阶紧致差分格式具有高分辨率,更适合求解小尺度波动问题.数值计算结果也表明：虽然优化格式仍然是四阶精度,但是要比传统四阶紧致差分格式的计算误差小；尤其对于小尺度波动,优化格式的计算误差会更小.对于行波问题,优化格式能够更加准确的模拟波动的传播行为,其优势也更加明显.理论分析和数值算例的比较结果均表明,优化的紧致差分格式更适合求解小尺度波动问题.更多还原

【Abstract】 The research and application of compact fnite diference scheme with high order ofaccuracy have received much attentions in recent years. This kind of scheme plays impor-tant roles in scientifc computations. As the increasing complexity scientifc computations,the requirement on the numerical scheme is increasing also. Thus, the numerical schemeswith lower order do not fulfll the requirement. As a result, it is necessary to develop highorder and high accuracy numerical scheme.The content of this thesis can be ascribed to the following three points:First, based on the idea of diference schemes pseudo-wavenumbers should resolvethe physical wavenumbers as accurate high as possible we proposed an optimal tridiagonalcompact fnite diference scheme with fourth order accuracy and high resolution. On the onehand, the optimal compact scheme can be solved efciently by the algorithms which we havedeveloped recently to solve the (cyclic) tridiagonal equations. Thus, the frst derivation ofpartial diferential equations can be solved efciently by the optimized compact diferencescheme efcienty; On the other hand, the maximum of pseudo-wavenumber can be resolvedby the optimal scheme is2.5761, and larger than the pseudo-wavenumber of1.13097,which can be resolved by the traditional fourth order compact schemes. Therefore, theoptimized compact diference scheme is more appropriate to resolve small scale waves influid dynamics and aero-acoustic dynamics.Second, comparisons among the optimized fourth order compact fnite diference scheme, sixth and eighth order compact diference are performed. The comparative results showedthat the optimized fourth order scheme with low accuracy has higher resolution than thatof the sixth and eight order schemes. Therefore, we have to bear in mind that the accuracyand the resolution are two diferent concepts and should not be replaced with each other.In contrast, one should choose a scheme with appropriate order of accuracy and resolutionaccording to the problem needed to be solved.Third, compared with traditional fourth-order compact fnite diference scheme, theoptimized fourth order compact fnite diference scheme has high resolution and is moresuitable to resolve small scale fluctuation problems in practical applications. Numerical ex-periments illustrated that: although the accuracy of the optimized scheme is fourth-order,it has a smaller error than that of the traditional fourth-order compact fnite diference,especially for the small scale fluctuations. Our numerical experiments also showed theadvantages of the optimized fourth-order compact fnite diference scheme in simulatingthe wave propagation problem. Thus, according to the theoretical analysis and numericalexperiment results, we conclude that optimized compact fnite diference scheme is moreappropriate to resolve small scale fluctuations problems.更多还原