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

Anindya Goswami, Kuldip Singh Patel, Pradeep Kumar Sahu

TL;DR

This work addresses the stability of fully discrete, fourth-order compact schemes for variable-coefficient convection–diffusion PDEs. It introduces a difference-equation based spectral analysis that reduces stability to the roots of a characteristic polynomial $D^1_N$ derived from the amplification matrix $W = X^{-1}Y$, and demonstrates a sufficient condition for unconditional stability. The key contributions include the first stability analysis for such fully discrete compact schemes with variable coefficients, the establishment of unconditional stability for the constant-coefficient limit, and a practical bound on the condition number of the amplification matrix, supported by numerical experiments. The results provide a rigorous foundation for robust high-order compact schemes in variable-coefficient settings and offer pathways to generalize the approach to multi-dimensional problems and systems.

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.

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

TL;DR

This work addresses the stability of fully discrete, fourth-order compact schemes for variable-coefficient convection–diffusion PDEs. It introduces a difference-equation based spectral analysis that reduces stability to the roots of a characteristic polynomial derived from the amplification matrix , and demonstrates a sufficient condition for unconditional stability. The key contributions include the first stability analysis for such fully discrete compact schemes with variable coefficients, the establishment of unconditional stability for the constant-coefficient limit, and a practical bound on the condition number of the amplification matrix, supported by numerical experiments. The results provide a rigorous foundation for robust high-order compact schemes in variable-coefficient settings and offer pathways to generalize the approach to multi-dimensional problems and systems.

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.
Paper Structure (14 sections, 9 theorems, 77 equations, 4 tables)

This paper contains 14 sections, 9 theorems, 77 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Stability analysis for variable coefficient problem
  3. Backward Euler compact scheme
  4. Crank-Nicolson compact scheme
  5. Special Case: Constant coefficient problem
  6. Backward Euler compact scheme
  7. Crank Nicolson compact scheme
  8. Condition Number
  9. Numerical Illustrations
  10. Conclusions and future directions
  11. MATLAB codes
  12. Expression of $D^1_N$
  13. Roots of $D^1_N$
  14. Positivity test of roots of $D^1_N$

Key Result

Lemma 2

For any fixed $\lambda\in \mathbb{C}$, and $N\ge 2$, let the vector $x= (x_0,x_1,\dots,x_N)\in \mathbb{C}^{N+1}$ solve linear_system_FB_v with $x_0=1$. Then where $\mathscr{A}_N$ is a collection of finite sequences $s:=\{s_j\}_{j=1}^{N-1}$ such that and

Theorems & Definitions (25)

  • Remark 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Remark 4
  • Theorem 5
  • proof
  • Remark 6
  • Remark 7
  • ...and 15 more