中文摘要：

本文在非均匀网格上给出了求解非定常对流扩散方程的一种高精度紧致差分格式,特别适合边界层和大梯度等问题的求解.从稳态对流扩散方程入手,首先,基于非均匀网格上的泰勒级数展开对空间导数项进行离散,然后对时间项采用二阶向后欧拉差分公式,从而得到一维非定常对流扩散方程在非均匀网格上的三层全隐式紧致差分格式.新格式在时间具有二阶精度,空间具有三到四阶精度,并且是无条件稳定的.最后,通过数值实验验证了本文格式的精确性,以及在处理诸如边界层和大梯度问题上的优势.

英文摘要：

A high accuracy compact finite difference scheme with non-uniform grids is proposed to solve unsteady convection diffusion equations,which are used to describe boundary layer problems or locally large gradient problems,etc.The new method starts from the discretization of the steady convection diffusion equation.Firstly,the spatial derivatives are discretized by using the Taylor series expansion on non-uniform grids.Then,the second order backward Eulerian difference formula is used to discretize the temporal derivative term.The three-level full implicit compact difference scheme on non-uniform grids for solving the onedimensional unsteady convection diffusion equation is derived.The new scheme has the second order accuracy in time and the third to fourth order accuracy in space and is unconditionally stable.Finally,some numerical experiments are conducted to demonstrate the high accuracy and the advantages in solving boundary layer problems or locally large gradient problems.