变系数微分方程的谱方法研究

It is important to solve linear and nonlinear differential problems with variable coefficients both in mathematical theory and physics simulation. This project will study the solution of nonlinear ordinary differential equations with variable coefficients via spectral methods. In additional, two dimensional separable elliptic partial differential equations with variable coefficients are solved and applied to equations of mathematical physics containing variable coefficients. By constructing differential operators, product operators and conversion operators between Chebyshev polynomials and ultraspherical polynomials, the differential equations are discretized in frequency space. For nonlinear differential equations, the nonlinear term is calculated via discrete Chebyshev transform, we solve the nonlinear equations using the Newton’s method. For two-dimensional variable coefficient problem , we construct the basis functions which satisfy the boundary conditions to represent the numerical solution , calculate the product matrix of bivariate functions, solve linear equations directly with the cost O(m^2 N^2), where m is bandwidth, in contrast to the cost O(N^6) in collocation methods. We further solve Navier-Stokes equations with variable density and make spectral method more popular.

研究变系数线性、非线性微分问题，无论在数学理论研究还是在物理过程模拟中都有极其重要的意义。本项目将研究解变系数非线性常微分方程与二维变系数变量可分的椭圆型偏微分方程的谱算法，应用于含有变系数数学物理方程的数值计算中。通过构造简单的微分算子、乘积算子和Chebyshev多项式与Ultrasoherical多项式之间的转换算子，在谱系数空间离散微分方程。对于非线性微分方程，我们采用离散的Chebyshev变换计算非线性项，用Newton方法迭代求解非线性方程组。对于二维变系数问题，我们用满足边值条件的基函数表示数值解，计算双变量函数的乘积矩阵，直接解带状的线性方程组，其计算复杂性为O(m^2 N^2), 其中m是矩阵的带宽，优于配置方法解二维微分方程的O(N^6)运算量。以此为基础，我们进一步解变密度的Navier-Stokes方程组。项目的研究将使谱方法应用于更广泛的问题求解中。

项目研究二维、三维椭圆型偏微分方程的高效最优谱算法。针对二维直角坐标区域上不同的边值条件，如周期边界，Dirichlet边界，Neumann边界，Robin边界条件等，对典型的椭圆型偏微分方程，如Poisson方程、Helmholtz方程提出高效快速求解的谱系数展开方法，并应用于抛物型偏微分方程的空间变量数值离散计算。通过交替方向隐式迭代求解Sylvester矩阵方程，能得到如同有限差分五点格式计算Poisson方程的快速谱算法。基于Poisson方程的最优谱算法，我们进一步研究了变系数和非线性的偏微分方程快速谱算法，应用于不可压缩流体的Navier-Stokes方程组，以及相场模型。项目的研究将使谱方法应用于更广泛的问题求解中。