时空耦合谱元方法求解一维Burgers方程

Time-Space Coupled Spectral Element Method for One-Dimensional Burgers Equation

针对Burgers方程在空间离散格式与时间离散格式方面的精度匹配问题,提出了一种时空耦合谱元方法,求解了一维Burgers方程。求解时在时间及空间方向同时采用了谱元方法离散方程,推导了求解过程,比较了空间方向采用谱元离散结合时间方向分别采用向后欧拉方法、四阶Runge-Kutta方法和四阶Adams-Bashforth方法的数值精度以及时空匹配特性,研究了时间方向网格单元数及插值节点数对时空耦合谱元方法数值精度的影响。研究显示:时空耦合谱元方法能够求解Burgers方程且与传统的时间差分方法相比能够获得更高的数值精度;随着空间方向单元内插值阶数的不断增大,时空耦合谱元方法的数值精度不断提高,且保留了指数阶收敛的特点,具有较好的时空匹配特性;当空间网格划分方式固定时,时间方向上增加单元数或单元内插值阶数,对数值精度提高影响不大,这一结论与相关研究结果一致。研究内容对引入与空间谱元方法精度相匹配的时间离散格式具有指导意义。

To improve the matching of numerical accuracy between time and space for Burgers equation,a time-space coupled spectral element method is proposed to solve one-dimensional Burgers equation.The equation is discretized with the spectral element method in both time and space directions,and detailed derivation is displayed.Additionally,the numerical accuracy and the matching characteristic of time and space of the proposed method are compared with those of three other schemes in which spectral element discretization is adopted in space direction while the backward Euler method,the fourth-order Runge-Kutta method and the fourth-order AdamsBashforth method are employed,respectively,in time direction.The influences of the numbers of elements and interpolation nodes on the numerical accuracy in time direction are investigated.It is verified that：1）higher accuracy can be obtained with the proposed time-space coupled spectral element method compared with conventional temporal difference schemes in time direction;the numerical accuracy is continuously promoted with the increasing interpolation order,and the exponential convergence rate is still maintained,which shows a better characteristic of matchingbetween time and space;and 3）increasing the elements or interpolation nodes in time direction with fixed space meshing has slight effect on improvement of the numerical accuracy,which coincides with the related research in literature.