Acceleration of the Jacobi iterative method by factors exceeding 100 using scheduled relaxation

Xiyang I.A. Yang; Rajat Mittal

doi:10.1016/j.jcp.2014.06.010

Acceleration of the Jacobi iterative method by factors exceeding 100 using scheduled relaxation

Xiyang I.A. Yang, Rajat Mittal

Mechanical Engineering

Research output: Contribution to journal › Article › peer-review

58 Scopus citations

Abstract

We present a methodology that accelerates the classical Jacobi iterative method by factors exceeding 100 when applied to the finite-difference approximation of elliptic equations on large grids. The method is based on a schedule of over- and under-relaxations that preserves the essential simplicity of the Jacobi method. Mathematical conditions that maximize the convergence rate are derived and optimal schemes identified. The convergence rate predicted from the analysis is validated via numerical experiments. The substantial acceleration of the Jacobi method enabled by the current method has the potential to significantly accelerate large-scale simulations in computational mechanics, as well as other areas where elliptic equations are prominent.

Original language	English (US)
Pages (from-to)	695-708
Number of pages	14
Journal	Journal of Computational Physics
Volume	274
DOIs	https://doi.org/10.1016/j.jcp.2014.06.010
State	Published - Oct 1 2014

All Science Journal Classification (ASJC) codes

Numerical Analysis
Modeling and Simulation
Physics and Astronomy (miscellaneous)
General Physics and Astronomy
Computer Science Applications
Computational Mathematics
Applied Mathematics

Access to Document

10.1016/j.jcp.2014.06.010

Cite this

@article{e00aa7e1a8b64936b56326c3eaef60a5,

title = "Acceleration of the Jacobi iterative method by factors exceeding 100 using scheduled relaxation",

abstract = "We present a methodology that accelerates the classical Jacobi iterative method by factors exceeding 100 when applied to the finite-difference approximation of elliptic equations on large grids. The method is based on a schedule of over- and under-relaxations that preserves the essential simplicity of the Jacobi method. Mathematical conditions that maximize the convergence rate are derived and optimal schemes identified. The convergence rate predicted from the analysis is validated via numerical experiments. The substantial acceleration of the Jacobi method enabled by the current method has the potential to significantly accelerate large-scale simulations in computational mechanics, as well as other areas where elliptic equations are prominent.",

author = "Yang, {Xiyang I.A.} and Rajat Mittal",

note = "Funding Information: XY was supported by Office of Naval Research grant N00014-12-1-0582 . RM would like to acknowledge support from Office of Naval Research as well as NSF (grants IOS1124804 , IIS1344772 and Office of International Science and Engineering 1243482 ). ",

year = "2014",

month = oct,

day = "1",

doi = "10.1016/j.jcp.2014.06.010",

language = "English (US)",

volume = "274",

pages = "695--708",

journal = "Journal of Computational Physics",

issn = "0021-9991",

publisher = "Academic Press Inc.",

}

TY - JOUR

T1 - Acceleration of the Jacobi iterative method by factors exceeding 100 using scheduled relaxation

AU - Yang, Xiyang I.A.

AU - Mittal, Rajat

N1 - Funding Information: XY was supported by Office of Naval Research grant N00014-12-1-0582 . RM would like to acknowledge support from Office of Naval Research as well as NSF (grants IOS1124804 , IIS1344772 and Office of International Science and Engineering 1243482 ).

PY - 2014/10/1

Y1 - 2014/10/1

N2 - We present a methodology that accelerates the classical Jacobi iterative method by factors exceeding 100 when applied to the finite-difference approximation of elliptic equations on large grids. The method is based on a schedule of over- and under-relaxations that preserves the essential simplicity of the Jacobi method. Mathematical conditions that maximize the convergence rate are derived and optimal schemes identified. The convergence rate predicted from the analysis is validated via numerical experiments. The substantial acceleration of the Jacobi method enabled by the current method has the potential to significantly accelerate large-scale simulations in computational mechanics, as well as other areas where elliptic equations are prominent.

AB - We present a methodology that accelerates the classical Jacobi iterative method by factors exceeding 100 when applied to the finite-difference approximation of elliptic equations on large grids. The method is based on a schedule of over- and under-relaxations that preserves the essential simplicity of the Jacobi method. Mathematical conditions that maximize the convergence rate are derived and optimal schemes identified. The convergence rate predicted from the analysis is validated via numerical experiments. The substantial acceleration of the Jacobi method enabled by the current method has the potential to significantly accelerate large-scale simulations in computational mechanics, as well as other areas where elliptic equations are prominent.

UR - http://www.scopus.com/inward/record.url?scp=84903889423&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84903889423&partnerID=8YFLogxK

U2 - 10.1016/j.jcp.2014.06.010

DO - 10.1016/j.jcp.2014.06.010

M3 - Article

AN - SCOPUS:84903889423

SN - 0021-9991

VL - 274

SP - 695

EP - 708

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -

Acceleration of the Jacobi iterative method by factors exceeding 100 using scheduled relaxation

Abstract

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Fingerprint

Cite this