三对角四阶跳点紧致格式优化和初步应用

An Optimized Fourth-order Staggered Tridiagonal Compact Scheme and Its Applications

根据修正波数应在充分大的波数范围内接近准确波数的思想,构造了优化的3对角4阶跳点紧致差分格式及插值格式。优化跳点紧致格式仍然具有4阶精度,但提高了分辨率,能够在更大的波数范围内保持群速度特性。通过实验数据表明,优化跳点紧致差分格式的分辨率可达到0.86π,优化紧致插值格式可达0.63π,可较好保持群速度的最大波数为0.75π,均大于标准4阶和6阶跳点紧致格式。分别用优化格式,标准4阶和6阶跳点紧致格式计算小尺度波动的性能,结果表明优化格式在模拟小尺度波动,在减小计算误差并保持群速度方面具有明显优势。

Based on the idea that the modified wavenumbers should be as close to the exact wavenumbers as possible, an optimal tridiagonal fourth-order compact difference scheme and an interpolation scheme on the staggered grid system are proposed in this paper. Although its accuracy is of the 4th order, the optimal scheme enjoys a high resolution and at the same time preserves the characteristics of the group velocity. The numerical calculations show that the maximum resolvable wavcnumbers obtained with the optimal compact difference （interpolation） scheme is 0.86π （0.63π）. The group velocity can be preserved for the wavenumber less than 0.75π. All these values are better than those obtained from the standard compact schemes of fourth or sixth order. Finally, the optimal scheme, the standard fourth order compact scheme and the sixth order compact schemes are employed to calculate the first derivation and the propagations of small scale waves. The results show that the optimal scheme is superior to other two schemes with respect to the reduction of errors and the preservation of the group velocity.