极坐标系下Fourier-Legendre谱元方法与有限差分法数值扩散的比较

提出一种Fourier-Legendre谱元方法用于求解极坐标系下的Navier-Stokes方程,其中极点所在单元的径向采用Gauss-Radau积分点,避免了r=0处的1/r坐标奇异性。时间离散采用时间分裂法,引入数值同位素模型跟踪同位素的输运过程验证数值模拟的精度,分别利用谱元法和有限差分法的迎风差分格式求解匀速和加速坩埚旋转流动中的同位素方程。计算结果表明,有限差分法中的一阶迎风差分格式存在严重的数值假扩散,二阶迎风差分格式的数值结果较精确,增加节点可以有效地缓解数值扩散。然而,谱元法具有以较少节点得到高精度解的优势。

Comparison of the Fourier-Legendre spectral element method and the finite difference method on the numerical diffusion in polar coordinate

For solving the Navier-Stokes equations in polar coordinates,a Fourier-Legendre spectral element method (SEM) with Gauss-Radau quadrature points in the radical direction for the element involving the origin r=0 is proposed to avoid the coordinate singularity 1/r.The time-splitting method is employed in the temporal discretization.A numerical isotope model is applied to track the isotope transport and therefore to evaluate the accuracy of numerical simulation.The isotope equation is solved by both the SEM and a finite-difference method (FDM) with an upwind difference scheme in the flows driven by a steady and an accelerated crucible rotations.The results demonstrate that a severe false numerical diffusion appears in FDM with the first-order upwind difference scheme.The accuracy of the second-order upwind scheme is higher than that of the first-order upwind scheme,and a dense mesh is helpful to alleviate the false diffusion effectively.However,the SEM exhibits an obvious advantage to achieve the high accuracy solution with even relative fewer nodes.