对流扩散方程的非振荡守恒特征差分法

Non Oscillation Conservative Characteristic Difference Method for Solving Convection Diffusion Equations

摘要: 当求解对流占优扩散问题时，采用传统的二次拉格朗日插值的特征差分法，会出现数值振荡，且不满足质量守恒。结合算子分裂，本质非震荡和MMOCAA质量校正，提出了求解对流扩散方程的非震荡的守恒特征差分法。首先采用局部一维(LOD)法把一个二维偏微分方程分裂成x方向和y方向的两个一维的偏微分方程组；其次在每个方向上利用二阶本质非振荡和MMOCAA格式进行数值计算。数值实验验证格式满足非震荡和质量守恒，能够有效地解决大型对流占优扩散问题。

Abstract: The characteristic difference method based on quadratic Lagrange interpolation is used to solve the convection dominated diffusion problem, but there will be a larger numerical oscillation and not conservative. Combining the operator splitting, non-oscillatory and mass correction methods, the non-oscillation conservative characteristic difference methods are proposed to solve the convection diffusion equations. Firstly, the partial differential equations in two dimensions are splitting into two one-dimensional partial differential equations along the x-direction and the y-direction, respectively. Secondly, the second-order essentially non-oscillation and MMOCAA schemes are presented to compute the equations. By the numerical results, it shows that the scheme not only meets non-oscillatory and mass conservation, but also effectively solves the convection-dominant diffusion problems.