一种求解对流扩散方程的无条件稳定算法  

An unconditionally stable scheme for solving convection-diffusion equation

作　　者：张迪[1] 缪小平[1] 彭福胜[1] 江丰[1] 魏子杰[1] 

机构地区：[1]解放军理工大学国防工程学院,南京210007

出　　处：《水动力学研究与进展（A辑）》2017年第2期158-164,共7页Chinese Journal of Hydrodynamics

基　　金：江苏省自然科学基金(BK20131067)~~

摘　　要：该文提出了一种求解二维对流扩散方程的无条件稳定算法。该算法将方程的时间项通过加权拉盖尔多项式作为正交基函数进行展开,利用Galerkin原理消除时间变量,导出隐式差分方程,并通过所得到的展开系数重构速度场或温度场等数值结果,从而突破了传统显式差分格式稳定性条件的限制,实现求解过程无条件稳定。为评价该算法的精度与效率,设计了两个数值算例,并将其与传统的显式差分格式和交替方向隐式差分格式进行了对比分析。结果表明:算法的精度与时间步长无关,对求解含有精细结构的对流扩散问题具有明显的效率优势。An unconditionally stable method for solving the convection-diffusion equation is proposed in this work. The time derivatives in the equation are expanded by the weighted Laguerre functions, by applying a temporal Galerkin＇s testing procedure to eliminate the time variable, an implicit difference equation can be derivated under no convergent conditions. The expanded coefficients of the field can be obtained by solving the equation of sparse matrix recursively and an unconditionally stable scheme is constructed. Two numerical experiments are conducted to validate the accuracy and efficiency of the present method. The numerical results have shown that the precision of this proposed method is independent of the time step, and the method is efficient when it comes to a computational domain with a thin layer or fine structure in convection-diffusion problems.

关 键 词：拉盖尔多项式 无条件稳定算法 有限差分法 对流扩散方程 

分 类 号：O351[理学—流体力学]