针对现有的各种数值方法在求解一维对流扩散方程时容易出现的数值振荡、假扩散等计算稳定性和计算精度不足问题,提出应用独立覆盖流形法进行数值求解的新思路,即分区的多项式级数逼近。基于标准的伽辽金法推导一维对流扩散方程的独立覆盖流形法求解公式。采用场变量的一阶导数在独立覆盖之间的窄条形覆盖重叠区域是否连续的后验误差估计方法,通过覆盖加密和级数升阶的h-p型混合自适应进行自动求解。给出的稳态和非稳态分析算例结果表明:分区级数的数值解稳定地逼近于精确解,最终两者很好地吻合;对于对流占优问题,自适应求解可以有效避免数值振荡。另外还尝试了将数值解代回微分方程计算残差作为误差指标,如果能使微分方程逐点满足,那么将是对数值解最严格的误差判断。
Abstract
In solving one-dimensional convection-diffusion equations, present numerical methods are prone to suffer from stability and accuracy problems caused by numerical oscillation and pseudo-diffusion. In view of this, an idea of applying Numerical Manifold Method (NMM) based on independent covers (the approximation using polynomial series piecewise-defined) to the numerical solution is proposed. The solution formula of the one-dimensional convection-diffusion equation is derived based on the standard Galerkin method. The posterior error estimation method about the continuity of the first-order derivative of field variable in the narrow overlapping area between independent covers is used for the automatic solving by h-p hybrid self-adaptive analysis with mesh refinement and ascending series order. The results of the steady-state and unsteady-state analysis examples show that the numerical solution of the piecewise-defined series steadily approximates and finally well fits the exact solution. For the convection-dominated problem, the adaptive solution effectively avoids numerical oscillation. In addition, the error index of the residual by substituting the numerical result back to the differential equation is successfully attempted. If the differential equation is solved point by point, the method would the most stringent error judgment for the numerical solution so far.
关键词
对流扩散方程 /
数值振荡 /
自适应分析 /
数值流形法 /
独立覆盖
Key words
convection-diffusion equation /
numerical oscillation /
self-adaptive analysis /
Numerical Manifold Method (NMM) /
independent covers
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