Numerical Methods for the Compressible Navier-Stokes Equations Using an Explicit Time-Marching Technique: (Comparison between the Four Stage Runge-Kutta and Two Stage Rational Runge-Kutta Schemes)

Masato Furukawa, Hidetoshi Tomioka, Masahiro Inoue

Research output: Contribution to journalArticle

Abstract

The four stage Runge-Kutta(4SRK) and two stage Rational Runge-Kutta(2SRRK) schemes have been applied to solve the compressible Navier-Stokes equations, in order to develop a fast, accurate explicit time-marching technique suitable for vectorization. On a supercomputer, a problem of shock-boundary layer interaction is calculated by use of these schemes combined with local timestepping. It is shown that the 4SRK scheme with the fourth-order central differencing of convective terms is stable out to a Courant number of 2.06 according to Neumann's stability criterion. The practical limit of Courant number has been very close to the theoretical limit. At a high Reynolds number, the Courant number limit of 2SRRK scheme is significantly less than that of conventional explicit methods. Both the 4SRK and 2SRRK schemes are readily vectorizable. The use of implicit residual averaging reduces iterations, but is not suitable for vectorization.

Original languageEnglish
Pages (from-to)3874-3879
Number of pages6
JournalTransactions of the Japan Society of Mechanical Engineers Series B
Volume52
Issue number484
DOIs
Publication statusPublished - Jan 1 1986

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time marching
Supercomputers
Stability criteria
Navier-Stokes equation
Navier Stokes equations
Numerical methods
Boundary layers
Reynolds number
supercomputers
high Reynolds number
iteration
boundary layers
shock
interactions

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanical Engineering

Cite this

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title = "Numerical Methods for the Compressible Navier-Stokes Equations Using an Explicit Time-Marching Technique: (Comparison between the Four Stage Runge-Kutta and Two Stage Rational Runge-Kutta Schemes)",
abstract = "The four stage Runge-Kutta(4SRK) and two stage Rational Runge-Kutta(2SRRK) schemes have been applied to solve the compressible Navier-Stokes equations, in order to develop a fast, accurate explicit time-marching technique suitable for vectorization. On a supercomputer, a problem of shock-boundary layer interaction is calculated by use of these schemes combined with local timestepping. It is shown that the 4SRK scheme with the fourth-order central differencing of convective terms is stable out to a Courant number of 2.06 according to Neumann's stability criterion. The practical limit of Courant number has been very close to the theoretical limit. At a high Reynolds number, the Courant number limit of 2SRRK scheme is significantly less than that of conventional explicit methods. Both the 4SRK and 2SRRK schemes are readily vectorizable. The use of implicit residual averaging reduces iterations, but is not suitable for vectorization.",
author = "Masato Furukawa and Hidetoshi Tomioka and Masahiro Inoue",
year = "1986",
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