论文标题
抛物线PDE的紧凑型方案的矩阵方法稳定性和鲁棒性
Matrix method stability and robustness of compact schemes for parabolic PDEs
论文作者
论文摘要
考虑对流扩散方程的完全离散问题。它包含用于空间离散化的紧凑型近似值，以及用于时间离散化的曲柄 - 尼科森方案。已经应用了Tridiagonal Toeplitz矩阵和Gerschgorin Circle定理的倒数条目的表达式来定位放大矩阵的特征值。得出相关矩阵的条件编号上的上限。它显示为$ \ MATHCAL {o} \ left（\ frac {ΔV} {Δz^2} \ right）$，其中$ΔV$和$Δz$分别是时间和空间步进尺寸。已经添加了一些数值插图来补充理论发现。
The fully discrete problem for convection-diffusion equation is considered. It comprises compact approximations for spatial discretization, and Crank-Nicolson scheme for temporal discretization. The expressions for the entries of inverse of tridiagonal Toeplitz matrix, and Gerschgorin circle theorem have been applied to locate the eigenvalues of the amplification matrix. An upper bound on the condition number of a relevant matrix is derived. It is shown to be of order $\mathcal{O}\left(\frac{δv}{δz^2}\right)$, where $δv$ and $δz$ are time and space step sizes respectively. Some numerical illustrations have been added to complement the theoretical findings.
