三对角四阶紧致差分格式的优化和初步应用

 提高数值解的精度和分辨率,有助于更精确地求解日趋复杂的工程问题。本文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的具有高分辨率的三对角紧致差分格式。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。数值计算结果表明：(1) 虽然优化格式仍然是四阶精度,但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;(2) 对于行波问题,优化格式能够更加准确地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。

Optimization of Fourth-order Compact Finite Difference Triangular Scheme and Its Initial Applications

In order to improve the precision and resolution of a numerical scheme used to solve the complex scientific and engineering problems, it is necessary for difference scheme to resolve wave numbers with as high precision as possible. Based on this idea, a triangular compact finite difference scheme with fourth-order accuracy and high resolution is proposed. On one hand, this compact scheme could be efficiently solved by the algorithms which are recently developed to solve the (cyclic) triangular equations, therefore, the first derivation could be efficiently calculated by the optimized compact difference scheme with higher resolution and less amount of calculation; on the other hand, it has a maximum accuracy wave number of 2.5761, comparing with that of 1.13097 by using traditional schemes. In short, the optimized compact difference scheme is more appropriate to simulate small scale fluctuations in fluid dynamics. Numerical computation experiments illustrate that(1) even though the optimized scheme is still fourth-order, it has a smaller error than that of traditional fourth-order compact finite difference, especially for the small scale fluctuations; (2) for the problems involving traveling wave, the optimized scheme is able to simulate wave propagation behavior more accurately. Both the theoretical analysis and numerical experiments indicate that the optimized compact finite difference scheme is more appropriate to resolve the problems with small scale fluctuations.