变对流系数非定常对流扩散方程的一类新的隐式四阶紧致格式
A new family of implicit fourth order compact schemes for unsteady convection-diffusion equation with variable convection coefficient
AI总结：
针对变对流系数非定常对流扩散方程，提出一类基于五点点模板、常数系数的隐式紧致有限差分格式，空间四阶精度，时间二阶或更低，具有对角占优、无条件稳定、高精度和良好相位误差特性，能有效求解线性和非线性问题。
摘要：
本文提出了一类新的隐式紧致有限差分格式，用于计算具有变对流系数的非定常对流扩散方程。这些格式在空间上具有四阶精度，在时间上根据加权时间平均参数的选择具有二阶或更低阶精度。所提出的格式将输运变量及其一阶导数作为未知量，结合了紧致离散化和Padé格式用于空间导数的优点。这些基于五点点模板和常数系数的格式，称为\emph{(5,5)常数系数四阶紧致} [(5,5)CC-4OC]，产生对角占优的方程组，并且比一些标准方法具有更高的精度和更好的相位及振幅误差特性。这些格式能够使用非单位网格纵横比，并且无条件稳定。它们能有效捕捉具有Dirichlet和Neumann边界条件的线性和非线性对流扩散方程的瞬态和稳态解。所提出的格式易于实现，除了线性对流扩散方程外，还可应用于不可压缩Navier-Stokes方程控制的问题。在所有情况下，获得的结果与解析解和现有数值结果高度一致，证明了所提出格式的效率和精度。
英文摘要：
In this paper, a new family of implicit compact finite difference schemes for computation of unsteady convection-diffusion equation with variable convection coefficient is proposed. The schemes are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter. The proposed schemes, where transport variable and its first derivatives are carried as the unknowns, combine virtues of compact discretization and Padé scheme for spatial derivative. These schemes which are based on five point stencil with constant coefficients, named as \emph{(5,5) Constant Coefficient 4th Order Compact} [(5,5)CC-4OC], give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods. These schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable. They efficiently capture both transient and steady solutions of linear and nonlinear convection-diffusion equations with Dirichlet as well as Neumann boundary condition. The proposed schemes can be easily implemented and are applied to problems governed by incompressible Navier-Stokes equations apart from linear convection-diffusion equation. Results obtained are in excellent agreement with analytical and available numerical results in all cases, establishing efficiency and accuracy of the proposed scheme.
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