求解对流方程的高精度紧致差分格式及软件实现
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space. Secondly, expending the one dimensional equation at (xi , tn ) , a three-level compact difference
scheme for solving the one-dimensional convection equation is proposed. It is named in HOC2. These stabilities are obtained by the von Neumann method. The accuracy and the stability of the present scheme are validated by some numerical experiments. We can draw a conclusion that HOC1 is better than HOC2 in the stability and the accuracy.
最后, 将本文所推导的格式接入到“PHOEBESolver”[1]求解软件, 使得偏微分方程数值解 的相关学者更加方便地使用本文格式.
关键词: 对流方程; 高精度; 紧致差分格式; LOD 方法; 有限差分法
I
Abstract
Convection equations is a kind of partial differential equations. Therefore, solving these equations has very important theoretical and practical significance. This paper establishes high-order compact difference scheme for solving convection equations. First of all, assuming that the one
HOC1.该格式在时间和空间上均具有四阶精度.再将方程在 (xi , tn ) 处展开,得到一种求解一维
对流方程的三层高精度紧致差分格式 HOC2.采用 Von Neumann 方法分析了两种格式的稳定性. 然后通过几个具有精确解的数值算例进行数值验证,可以看出,本文提出的 HOC1 格式具有较好 的稳定性和精确性.
Then, For the two-dimensional and three-dimensional convection equations, using the LOD method to making the two-dimensional and three-dimensional problems split one-dimensional equations. The one-dimensional convection equations use taylor series expansion and correction for the third derivative in the truncation error remainder of the central difference scheme in discretization of time and space. We can establish some high-order compact LOD schemes for solving two-dimensional and three-dimensional convection equations. The stability are obtained by the von Neumann method.The accuracy and reliability of these schemes are validated by some numerical experiments.
dimensional equation is established at (xi , tn1/2 ) , Taylor series expansion Leabharlann Baidund correction for the third
derivative in the truncation error remainder of the central difference scheme are used for discretization of time and space. So a two-level implicit compact difference scheme for solving the one-dimensional convection equation is proposed. It is named in HOC1. it is the fourth-order accuracy in both time and
摘要
对流方程是一类重要的偏微分方程.因此,数值求解该类方程具有非常重要的理论价值和实
际意义.本文建立了求解对流方程的高阶紧致差分格式.首先,假设方程在 (xi , tn1/2 ) 点成立,将
方程在时间方向和空间方向上均采用泰勒级数展开及对截断误差余项中的三阶导数进行修正的 方法对时间和空间导数进行离散,得到一种求解一维对流方程的两层高精度紧致全隐格式
其次,针对二维、三维对流方程,利用局部一维化(LOD)方法分裂为一维问题进行求解.并将 分裂后的一维对流方程在时间和空间上均采用泰勒级数展开及对截断误差余项中的三阶导数进 行修正的方法对时间和空间导数进行离散,得到二维、三维对流方程的高精度紧致 LOD 格式, 运用 Von Neumann 方法分析了该格式的稳定性,通过数值算例验证了格式的精确性和可靠性.
Finally, these schemes deduced in this paper are integrated into the software of "PHOEBESolver", which makes it easier for scholars in numerical solutions of partial differential equations to use these schemes in this paper.
scheme for solving the one-dimensional convection equation is proposed. It is named in HOC2. These stabilities are obtained by the von Neumann method. The accuracy and the stability of the present scheme are validated by some numerical experiments. We can draw a conclusion that HOC1 is better than HOC2 in the stability and the accuracy.
最后, 将本文所推导的格式接入到“PHOEBESolver”[1]求解软件, 使得偏微分方程数值解 的相关学者更加方便地使用本文格式.
关键词: 对流方程; 高精度; 紧致差分格式; LOD 方法; 有限差分法
I
Abstract
Convection equations is a kind of partial differential equations. Therefore, solving these equations has very important theoretical and practical significance. This paper establishes high-order compact difference scheme for solving convection equations. First of all, assuming that the one
HOC1.该格式在时间和空间上均具有四阶精度.再将方程在 (xi , tn ) 处展开,得到一种求解一维
对流方程的三层高精度紧致差分格式 HOC2.采用 Von Neumann 方法分析了两种格式的稳定性. 然后通过几个具有精确解的数值算例进行数值验证,可以看出,本文提出的 HOC1 格式具有较好 的稳定性和精确性.
Then, For the two-dimensional and three-dimensional convection equations, using the LOD method to making the two-dimensional and three-dimensional problems split one-dimensional equations. The one-dimensional convection equations use taylor series expansion and correction for the third derivative in the truncation error remainder of the central difference scheme in discretization of time and space. We can establish some high-order compact LOD schemes for solving two-dimensional and three-dimensional convection equations. The stability are obtained by the von Neumann method.The accuracy and reliability of these schemes are validated by some numerical experiments.
dimensional equation is established at (xi , tn1/2 ) , Taylor series expansion Leabharlann Baidund correction for the third
derivative in the truncation error remainder of the central difference scheme are used for discretization of time and space. So a two-level implicit compact difference scheme for solving the one-dimensional convection equation is proposed. It is named in HOC1. it is the fourth-order accuracy in both time and
摘要
对流方程是一类重要的偏微分方程.因此,数值求解该类方程具有非常重要的理论价值和实
际意义.本文建立了求解对流方程的高阶紧致差分格式.首先,假设方程在 (xi , tn1/2 ) 点成立,将
方程在时间方向和空间方向上均采用泰勒级数展开及对截断误差余项中的三阶导数进行修正的 方法对时间和空间导数进行离散,得到一种求解一维对流方程的两层高精度紧致全隐格式
其次,针对二维、三维对流方程,利用局部一维化(LOD)方法分裂为一维问题进行求解.并将 分裂后的一维对流方程在时间和空间上均采用泰勒级数展开及对截断误差余项中的三阶导数进 行修正的方法对时间和空间导数进行离散,得到二维、三维对流方程的高精度紧致 LOD 格式, 运用 Von Neumann 方法分析了该格式的稳定性,通过数值算例验证了格式的精确性和可靠性.
Finally, these schemes deduced in this paper are integrated into the software of "PHOEBESolver", which makes it easier for scholars in numerical solutions of partial differential equations to use these schemes in this paper.