A novel difference equation approach for the stability and robustness of compact schemes for variable coefficient PDEs

TL;DR

This paper introduces a difference equation approach to analyze the stability and robustness of fourth-order compact schemes for variable coefficient PDEs, providing theoretical stability proofs and condition number estimates.

Contribution

It presents a new difference equation method for stability analysis and proves unconditional stability for constant coefficient cases of compact schemes.

Findings

Unconditional stability proved for constant coefficient problems.

Derived a sufficient stability condition for variable coefficient schemes.

Analyzed the condition number of the amplification matrix.

Abstract

Fourth-order accurate compact schemes for variable coefficient convection diffusion equations are considered. A sufficient condition for the stability of the fully discrete problem is derived using a difference equation based approach. The constant coefficient problems are considered as a special case, and the unconditional stability of compact schemes for such case is proved theoretically. The condition number of the amplification matrix is also analyzed, and an estimate for the same is derived. The examples are provided to support the assumption taken to assure stability.