Extensions and analysis of worst-case parameter in weighted Jacobi's method for solving second order implicit PDEs

Gregory J. Kimmel, Andreas Glatz

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

The optimal Jacobi parameter (ω) in Jacobi's iterative method is obtained for specific classes of matrices. We define ωopt as the worst-case optimal parameter. We show that matrices with nonzero elements only along the main diagonal and odd diagonals have ωopt=1. We show ωopt→1 holds for matrices with size n and nonzero diagonal d as n,d→∞, where d is the distance from the main diagonal. Finally, we show an application which exploits these derived properties to reduce the number of required Jacobi iterations. This is especially useful for physical problems that involve 2nd order implicit PDEs (e.g. diffusion, fluids) with large sparse matrices, where a change in discretization can change which diagonals are nonzero.

Original languageEnglish
Article number100003
JournalResults in Applied Mathematics
Volume1
DOIs
StatePublished - Jun 2019
Externally publishedYes

Keywords

  • Asymptotic analysis
  • Jacobi's method
  • PDE solvers
  • Weighted Jacobi's method

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