A Projection Method for the Conservative Discretizations of Parabolic Partial Differential Equations

Darae Jeong, Junseok Kim

Research output: Contribution to journalArticle

Abstract

We present a projection method for the conservative discretizations of parabolic partial differential equations. When we solve a system of discrete equations arising from the finite difference discretization of the PDE, we can use iterative algorithms such as conjugate gradient, generalized minimum residual, and multigrid methods. An iterative method is a numerical approach that generates a sequence of improved approximate solutions for a system of equations. We repeat the iterative algorithm until a numerical solution is within a specified tolerance. Therefore, even though the discretization is conservative, the actual numerical solution obtained from an iterative method is not conservative. We propose a simple projection method which projects the non-conservative numerical solution into a conservative one by using the original scheme. Numerical experiments demonstrate the proposed scheme does not degrade the accuracy of the original numerical scheme and it preserves the conservative quantity within rounding errors.

Original languageEnglish
Pages (from-to)1-18
Number of pages18
JournalJournal of Scientific Computing
DOIs
Publication statusAccepted/In press - 2017 Aug 17

Fingerprint

Parabolic Partial Differential Equations
Projection Method
Iterative methods
Partial differential equations
Discretization
Numerical Solution
Iterative Algorithm
Iteration
Conjugate Gradient
Rounding error
Multigrid Method
Discrete Equations
Experiments
Numerical Scheme
System of equations
Tolerance
Finite Difference
Approximate Solution
Numerical Experiment
Demonstrate

Keywords

  • Conservative discretization
  • Iterative methods
  • Projection method

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Software
  • Engineering(all)
  • Computational Theory and Mathematics

Cite this

@article{1d99620c59f64abdb6034297f4326a81,
title = "A Projection Method for the Conservative Discretizations of Parabolic Partial Differential Equations",
abstract = "We present a projection method for the conservative discretizations of parabolic partial differential equations. When we solve a system of discrete equations arising from the finite difference discretization of the PDE, we can use iterative algorithms such as conjugate gradient, generalized minimum residual, and multigrid methods. An iterative method is a numerical approach that generates a sequence of improved approximate solutions for a system of equations. We repeat the iterative algorithm until a numerical solution is within a specified tolerance. Therefore, even though the discretization is conservative, the actual numerical solution obtained from an iterative method is not conservative. We propose a simple projection method which projects the non-conservative numerical solution into a conservative one by using the original scheme. Numerical experiments demonstrate the proposed scheme does not degrade the accuracy of the original numerical scheme and it preserves the conservative quantity within rounding errors.",
keywords = "Conservative discretization, Iterative methods, Projection method",
author = "Darae Jeong and Junseok Kim",
year = "2017",
month = "8",
day = "17",
doi = "10.1007/s10915-017-0536-2",
language = "English",
pages = "1--18",
journal = "Journal of Scientific Computing",
issn = "0885-7474",
publisher = "Springer New York",

}

TY - JOUR

T1 - A Projection Method for the Conservative Discretizations of Parabolic Partial Differential Equations

AU - Jeong, Darae

AU - Kim, Junseok

PY - 2017/8/17

Y1 - 2017/8/17

N2 - We present a projection method for the conservative discretizations of parabolic partial differential equations. When we solve a system of discrete equations arising from the finite difference discretization of the PDE, we can use iterative algorithms such as conjugate gradient, generalized minimum residual, and multigrid methods. An iterative method is a numerical approach that generates a sequence of improved approximate solutions for a system of equations. We repeat the iterative algorithm until a numerical solution is within a specified tolerance. Therefore, even though the discretization is conservative, the actual numerical solution obtained from an iterative method is not conservative. We propose a simple projection method which projects the non-conservative numerical solution into a conservative one by using the original scheme. Numerical experiments demonstrate the proposed scheme does not degrade the accuracy of the original numerical scheme and it preserves the conservative quantity within rounding errors.

AB - We present a projection method for the conservative discretizations of parabolic partial differential equations. When we solve a system of discrete equations arising from the finite difference discretization of the PDE, we can use iterative algorithms such as conjugate gradient, generalized minimum residual, and multigrid methods. An iterative method is a numerical approach that generates a sequence of improved approximate solutions for a system of equations. We repeat the iterative algorithm until a numerical solution is within a specified tolerance. Therefore, even though the discretization is conservative, the actual numerical solution obtained from an iterative method is not conservative. We propose a simple projection method which projects the non-conservative numerical solution into a conservative one by using the original scheme. Numerical experiments demonstrate the proposed scheme does not degrade the accuracy of the original numerical scheme and it preserves the conservative quantity within rounding errors.

KW - Conservative discretization

KW - Iterative methods

KW - Projection method

UR - http://www.scopus.com/inward/record.url?scp=85027731588&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027731588&partnerID=8YFLogxK

U2 - 10.1007/s10915-017-0536-2

DO - 10.1007/s10915-017-0536-2

M3 - Article

SP - 1

EP - 18

JO - Journal of Scientific Computing

JF - Journal of Scientific Computing

SN - 0885-7474

ER -