一种求解反应对流扩散方程的稳定化谱元方法

摘要

针对谱元方法求解二维非稳态反应对流扩散方程中出现的稳定性问题,提出了一种稳定的高精度数值方法.该方法在空间上将Chebyshev谱元方法和一致逼近迎风方法相结合,时间上采用分步θ-格式.通过解析解算例验证了该方法的精度及数值稳定性,并对含有不同类型边界层的反应对流扩散问题进行了求解.研究表明:一致逼近迎风项的增加扩大了谱元方法求解反应对流扩散方程的稳定域,在对流项及反应项占优时保持了数值解的高精度;对于含有边界层的复杂反应对流扩散问题,数值解在整个计算区域内获得了一致收敛的结果.研究工作对谱元方法在反应对流扩散问题高精度数值求解中的应用提供参考.%A stabilized numerical method with high accuracy is proposed to solve the stability problems,which are generated by spectral element method for two-dimensional transient reaction-convection-diffusion equation.The method combines Chebyshev spectral element method with consistent approximate upwind method in space and adopts fractional-step θ-pattern in time.The numerical example is solved and compared with analytical solution to verify the accuracy and numerical stability.Reaction-convection-diffusion problems with different kinds of boundary layers are also solved.This approach reveals that the stability domain of the spectral element method for reaction-convection-diffusion equation is enlarged with the addition of the consistent approximate upwind term,and the high accuracy of the numerical solution is maintained when convection and reaction term are dominated.For the complicated reaction-convection-diffusion problems with boundary layers,the numerical solution is able to obtain uniform convergence in the whole computational domain.