谱元方法求解对流扩散方程及其稳定性分析

Spectral Element Method for Convection-Diffusion Equation with Stability Analysis

探讨了一维对流扩散方程的一种高精度数值解法,该解法在空间上采用了Chebyshev谱元方法,在时间上结合了半隐式Adams方法。通过数值算例验证了解法的可行性,利用特征分析法得到了对流扩散方程谱元求解时不同离散形式的稳定性条件,并对数值求解的稳定性进行了预测。通过时间步长、网格划分、对流项和黏性项插值阶数的影响研究表明:耦合Chebyshev谱元方法和半隐式Adams方法在求解对流扩散方程时能够获得高精度的数值解;时间离散时Adams方法的黏性项采用一阶插值形式、对流项采用二阶插值形式,在未增加计算量的同时能够获得较大的稳定区域和较高的计算精度。

The Chebyshev spectral element method combining with the semi-implicit Adams method is presented for solving the one-dimensional convection-diffusion equation, and the feasibililty is verified by a numerical example. The stability condition of different discrete forms of spectral element method is deduced with character analysis, and the influences of time step, grid partitioning, interpolation order of convective and viscous terms are discussed. It is demonstrated that numerical solution with high accuracy can be gained with the coupled spectral element and semi-implicit Adams method for the convection-diffusion equation. Larger stability domain and higher accuracy can be achieved without extra-calculation as first-order viscous terms and secondorder convective terms are interpolated in Adams method.