Matrix method stability and robustness of compact schemes for parabolic PDEs

TL;DR

This paper analyzes the stability and robustness of compact finite difference schemes for parabolic PDEs, focusing on eigenvalue location, condition number bounds, and numerical validation for convection-diffusion problems.

Contribution

It provides a novel matrix-based approach to assess stability and robustness of compact schemes for parabolic PDEs, including eigenvalue analysis and condition number estimates.

Findings

Eigenvalues of the amplification matrix are located using Gerschgorin circles.

An upper bound on the condition number is derived, showing dependence on discretization parameters.

Numerical results support the theoretical stability and robustness analysis.

Abstract

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 $\mathcal{O}\left(\frac{\delta v}{\delta z^2}\right)$, where $\delta v$ and $\delta z$ are time and space step sizes respectively. Some numerical illustrations have been added to complement the theoretical findings.