發(fā)布時(shí)間：2018-08-21 12:13

【摘要】：為研究基于Least-Squares變分及Galerkin變分兩種形式的譜元方法的求解特性,推導(dǎo)了極坐標(biāo)系中采用兩種變分方法求解環(huán)形區(qū)域內(nèi)Poisson方程時(shí)對(duì)應(yīng)的弱解形式,采用Chebyshev多項(xiàng)式構(gòu)造插值基函數(shù)進(jìn)行空間離散,得到兩種譜元方法對(duì)應(yīng)的代數(shù)方程組,由此分析了系數(shù)矩陣結(jié)構(gòu)的特點(diǎn)。數(shù)值計(jì)算結(jié)果顯示:Least-Squares譜元方法為實(shí)現(xiàn)方程的降階而引入新的求解變量,使得代數(shù)方程組形式更為復(fù)雜,但邊界條件的處理比Galerkin譜元方法更為簡(jiǎn)單;兩種譜元方法均能求解極坐標(biāo)系中的Poisson方程且能獲得高精度的數(shù)值解,二者絕對(duì)誤差分布基本一致;固定單元內(nèi)的插值階數(shù)時(shí),增加單元數(shù)可減小數(shù)值誤差,且表現(xiàn)出代數(shù)精度的特點(diǎn),誤差降低速度較慢,而固定單元數(shù)時(shí),在一定范圍內(nèi)數(shù)值誤差隨插值階數(shù)的增加而減小的速度更快,表現(xiàn)出譜精度的特點(diǎn);單元內(nèi)插值階數(shù)較高時(shí),代數(shù)方程組系數(shù)矩陣的條件數(shù)急劇增多,方程組呈現(xiàn)病態(tài),數(shù)值誤差增大,這一特點(diǎn)限制了單元內(nèi)插值階數(shù)的取值。研究?jī)?nèi)容對(duì)深入了解兩種譜元方法在極坐標(biāo)系中求解Poisson方程時(shí)的特點(diǎn)、進(jìn)一步采用相關(guān)分裂算法求解實(shí)際流動(dòng)問題具有參考價(jià)值。
[Abstract]:In order to study the characteristics of spectral element method based on Least-Squares variation and Galerkin variation, the corresponding weak solutions for solving Poisson equation in annular region by using two variational methods in polar coordinate system are derived. The interpolation basis function is constructed by Chebyshev polynomial and the algebraic equations corresponding to two spectral element methods are obtained. The characteristics of the structure of the coefficient matrix are analyzed. The numerical results show that the new solution variable is introduced to reduce the order of the equation by the method of the Least-Squares spectral element method, which makes the form of algebraic equations more complicated, but the boundary condition is simpler than the Galerkin spectral element method. The two spectral element methods can solve the Poisson equation in polar coordinate system and obtain high accuracy numerical solution. The absolute error distribution of the two methods is basically the same, and the numerical error can be reduced by increasing the number of elements when the order of interpolation is fixed. And it shows the characteristics of algebraic precision, the speed of error decreasing is slow, and when the number of elements is fixed, the numerical error decreases faster with the increase of interpolation order in a certain range, which shows the characteristic of spectral precision, and when the interpolation order is higher in the unit, the numerical error decreases more quickly with the increase of interpolation order in a certain range. The condition number of the coefficient matrix of algebraic equations increases sharply, the system of equations is ill-conditioned and the numerical error increases, which limits the value of interpolation order in the unit. The research is valuable for further understanding the characteristics of the two spectral element methods in polar coordinate system for solving the Poisson equation and further using the correlation splitting algorithm to solve the actual flow problem.
【作者單位】： 西安交通大學(xué)能源與動(dòng)力工程學(xué)院;
【基金】：國(guó)家重點(diǎn)基礎(chǔ)研究發(fā)展計(jì)劃資助項(xiàng)目(2012CB026004)
【分類號(hào)】：O241.82

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