OCS4格式的数值边界格式及其渐近稳定性

 根据多项式拟合数值边界格式(SFEBS)和Taylor展开数值边界格式(TEBS)相结合的思想,构造了与优化3对角4阶跳点紧致差分格式(OCS4)及其插值格式(OCI4)相匹配的具有4阶精度的数值边界格式(SF-TEBS4).通过计算格式特征值的理论分析表明,OCS4、OCI4格式在与数值边界格式SF-TEBS4格式相结合时,数值格式在整体上能够满足渐进稳定性的要求.一阶导数数值试验表明,OCS4、OCI4与4阶数值边界格式SF-TEBS4在数值模拟中相结合使用时,能够保证格式整体精度达到4阶,且计算误差较小;行波解数值模拟表明,这些格式的组合能够有效抑制数值计算的误差,具有能够长时间保持群速度和较强渐进稳定性的特性.理论分析和数值算例均表明,SF-TEBS4与OCS4和OCI4相结合,能够很好地求解小尺度波动问题.

A Numerical Boundary Scheme of OCS4 and Its Asymptotic Stability

Based on the ideas of the polynomial fitting numerical boundary scheme (SFEBS) and the Taylor expansion boundary scheme (TEBS), a fourth-order numerical boundary scheme (SF-TEBS4) is proposed in this paper. The SF-TEBS4 is an extension of the optimized fourth-order staggered tridiagonal compact difference scheme (OCS4) and the corresponding interpolation scheme (OCI4) on the staggered grid system, developed by the authors recently, for the equations with non-periodical physical boundary conditions. The asymptotic stability of the overall difference scheme, the combination of the numerical boundary scheme SF-TEBS4, and the inner points schemes OCS4 and OCI4, is analyzed. It is shown that, SF-TEBS4 combined with OCS4 and OCI4, can achieve the asymptotic stability. Moreover, the numerical experiment for determining the first order derivative of a function indicates that (1) the global accuracy of our scheme is fourth-order, and that (2) the computational error is reduced greatly. The numerical experiment for solving the wave propagation problem shows that the combination of SF-TEBS4 with OCS4 and OCI4 can effectively suppress the growth rate of the computational error, preserve the group velocity and the numerical asymptotic stability. The theoretical and numerical analyses show that the combination of SF-TEBS4 with OCS4 and OCI4 can be applied to simulate the propagation of small scale waves.