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上海市自然科学基金资助课题——谱方法研究,已由上海计算所副研究员郑家栋等经多年潜心研究获得重要理论结果,不久前在沪通过鉴定。 采用谱方法求解偏微分方程,国外在80年代才取得实质进展。由于它在计算时运用FFT技术,并且具有解越光滑、近似解精度高的优点,因而已被成功地应用于天气数值预报和非线性波、不可压缩流、可压缩流的计算。但是,如何处理被逼近解的奇异性和难于处理边界复杂的区域这两个问题,目前已成为谱方法研究的前沿课题。这项工作目前国内还未见报道,国外也刚起步。他们所取得的重要成果有:①对拟谱法解方程的收敛性作了园满的研究。国外Lamp的工作只论及L~2(Ω)空间的估计,且又不是最佳的。而他们的工作得到了Sobolev范数下最佳估计,其证明方法也不同于Lamp。②研究了闭曲线上的拟微分算子的拟谱配点方法,一大类重要的积微方程均可化为拟微分算子的方程,从而拓宽了谱方法的应用领域。③提出Fourier拟谱配点法,并用于实际问题。计算结果表明,能以较少的计算、存贮和配置点,取得比美籍华人肖家驹的Galerkin配点法更好的数值精确。④提出Chebyshev谱有限条法,用于分析平板弯曲,基函数可适应各种边界条件,计算结果与解析解相符。⑤设计了两个能在通用微机上执行的边界积分方程拟谱?
Shanghai Natural Science Foundation funded project - spectral method research, by the Shanghai Institute of Computing Associate Professor Zheng Jiadong after years of painstaking research to obtain important theoretical results, not long ago in Shanghai through the identification. Using spectral method to solve partial differential equations, foreign countries made substantial progress in the 1980s. Because it uses FFT technology in calculation, and has the advantages of more smooth solution and high approximate solution accuracy, it has been successfully applied to the calculation of weather numerical prediction and nonlinear wave, incompressible flow and compressible flow. However, how to deal with the singularity of the approximative solution and the difficulty of dealing with the complicated boundary areas have now become the forefront topics in spectral method research. At present, this work has not been reported domestically and abroad has just started. The important achievements they have made are as follows: (1) The convergence of the equations of the pseudospectral method has been thoroughly studied. The work of foreign lamps only deals with the estimation of L ~ 2 (Ω) space, which is not the best. And their work gets the best estimate under the Sobolev norm, which proves to be different from Lamp. (2) The quasi-spectral collocation method of the quasi-differential operators on the closed curve is studied. A large class of important microcoil equations can be transformed into the quasi-differential operator equations, which broaden the application fields of spectral methods. (3) Propose Fourier pseudo-matching point method and apply it to practical problems. The calculation results show that the numerical calculation is more accurate and accurate than the Galerkin collocation method proposed by Chinese Xiao Xiaoju with fewer calculation, storage and configuration points. ④ The Chebyshev spectrum finite strip method is proposed for the analysis of slab bending. The basis functions can be adapted to various boundary conditions. The calculated results agree well with the analytical solutions. ⑤ designed two can be implemented in a general computer boundary integral equation pseudo-spectrum?