摘要
将作者提出的数值摄动算法改进为区分离散单元内上游和下游并分别对通量进行高精度重构的双重数值摄动算法,与原(单重)摄动算法相比,双重摄动算法既提高了格式精度又明显扩大了格式的稳定域范围.利用双重摄动算法,即分别利用上游和下游基点变量的摄动重构将高阶流体力学关系及迎风机制耦合进二阶中心格式之中,由此构建了对流扩散方程的对网格Reynolds数的任意值均稳定(绝对稳定)高精度(四阶和八阶精度)三基点中心TVD差分格式,通过解析分析以及3个算例计算证实了构建格式的优良性能;3个算例包括一维线性、非线性(Burgers方程)和二维变系数对流扩散方程.数值计算表明:构建的格式在粗网格下不振荡,构建格式在粗网格时的最大误差L_∞和均方误差L_2与二阶中心格式在细网格时的相应误差一致,对线性方程,构建格式在细网格下可达到L_2精度阶.
In this paper the numerical-perturbation algorithm presented by the author is transformed from single perturbation reconstruction into dual one,in which the perturbation reconstruction of flux is performed by using respectively upstream and downstream nodes in a discrete element.Compared with the original single reconstruction using all nodes in the discrete element,dual perturbation reconstruction of flux can cut off propagation of convective anti-diffusion unstable information between upstream and downstream nodes and can couple both the high-order fluid dynamic relations and＂upwind biasing＂with the second order central difference scheme.Therefore,the accuracies of reconstructed schemes are raised and the stability range of reconstructed scheme is enlarged greatly.Two absolute stability,fourth- and eighth-order accurate central difference schemes（call them DPCS,for brevity） for the convective-diffusion equation are obtained.In the case of one dimension,DPCS are TVD schemes with order higher than second and they are nonoscillatory schemes for any values of grid Reynolds number.DPCS are reconstructed schemes of the classical second order central scheme coupling with both fluid dynamics effects and＂upwind biasing＂and do not introduce any artificial numerical dissipation.DPCS＇s excellent properties are proved by analyses and three computational examples, which include one-dimensional linear and nonlinear and two-dimensional convective-diffusion equations.As to calculation of Burgers equation,the well-known second order central difference scheme（2-CDS） oscillates and diverges on coarse grids,while the fourth- and eighth-order accurate DPCS do not oscillate.Both maximum error L_∞and mean square error L_2 of fourth- and eighth-order DPCS on coarse grids（grid number N=80,160） are approximately equal to those of 2CDS on fine grids（N=320）.From here we see that the present fourth- and eighth-order DPCS can capture discontinuities with high resolution.As to calculation of one-dimensional linear convective diffusion equation 2-CDS oscillates on coarse grids,while 4-DPCS and 8-DPCS do not oscillate;all 2-CDS,4-DPCS and 8-DPCS can reach to individual L_2 order-of-accurate on fine grids（grid number N≥320）; L_2 errors of 4-DPCS and 8-DPCS are greatly less than those of 2-CDS on fine grids.
出处
《力学学报》
EI
CSCD
北大核心
2010年第5期811-817,共7页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家自然科学基金资助项目(10872204)~~
关键词
计算流体力学
数值摄动算法
有限差分方法
对流扩散方程
computational fluid dynamics
finite difference method
numerical perturbation algorithm
convective-diffusion equation
