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Numerical Computation of Hyperbolic Conservation Laws Using Large Time Step Scheme with Artificial Compression Method

  • Research Article - Physics
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Abstract

Explicit formulation of one-dimensional Euler equation with TVD scheme is bounded by the stability criterion \(\hbox {CFL} <1\). This limitation does not allow large time steps and considerably increases simulation time. Large time step schemes have been developed for explicit formulations which are stable for larger CFL. TVD schemes are nonoscillatory but suffer with smearing of discontinuities. Artificial compression method (ACM) derived by Harten is appropriate to deal with smearing problem. Present work focuses this deficiency and present LTS TVD scheme results with ACM to investigate its behavior and consequences. One-dimensional shock tube problem for SOD, LAX, and inverse shock boundary conditions is solved to reveal the effect of ACM on large time step scheme at complex flow regions, namely expansion fan, contact, and shock wave. Results for all the three test cases with proposed methodology are satisfactory with some minor issues. Results depict that proper descriptions of parameters in ACM are very important, especially in the case of inverse shock boundary condition. It is concluded that large time step scheme with ACM effectively reduces numerical dissipation along with shorter simulation time. Such an approach for handling smearing problem in LTS TVD schemes proves its vital role in the design and analysis phase and provides more opportunities to examine in shorter time period. However, only one-dimensional problem is solved in present studies. Further studies should also be carried out with two-dimensional and three-dimensional test cases in order to reinforce currently computed results.

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Abbreviations

A :

Inviscid flux jacobian matrix

\({C}_{l (x)}\) :

Coefficient functions

D :

Small fractional constant

E :

Total energy

F :

Physical flux

K :

CFL restriction parameter

P :

ACM parameter

R :

Eigenvector matrix

\(R^{-1}\) :

Inverse eigenvector matrix

U :

Conservative variable vector

a :

Characteristic speed

c :

Speed of sound

f :

Numerical flux

g :

Flux correction

ğ:

Limiter function

i :

Grid pointer

:

ACM parameter

k :

Characteristic direction

m :

Number of eigenvalues

n :

Number of time steps taken

p :

Pressure

t :

Time

\(\Delta {t}\) :

Time step

u :

Velocity in x-direction

v :

Local CFL number

x :

Axial distance

\(\Delta {x}\) :

Grid spacing

\(\alpha \) :

Characteristic variable

\(\beta \) :

Numerical characteristic speed

\(\varepsilon \) :

Entropy fix parameter

\(\Phi \) :

Numerical dissipation term

\(\gamma \) :

Ratio of specific heat

\(\lambda \) :

Mesh ratio

\(\mu \) :

CFL parameter

\(\rho \) :

Density

\(\sigma \) :

Limiter function parameter

\(\psi \) :

Entropy correction function

\(\theta \) :

ACM switch

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Authors and Affiliations

  1. Institute of Space and Planetary Astrophysics, University of Karachi, Karachi, Pakistan

    Ihtram Ul Haq & Muhammad Jawed Iqbal

  2. Institute of Space Technology, Karachi, Pakistan

    Mukkarum Hussain

  3. Department of Mathematics, University of Karachi, Karachi, Pakistan

    Noor Fatima Siddiqui

Authors
  1. Ihtram Ul Haq
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  2. Mukkarum Hussain
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  3. Muhammad Jawed Iqbal
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  4. Noor Fatima Siddiqui
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Correspondence to Ihtram Ul Haq.

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Haq, I.U., Hussain, M., Iqbal, M.J. et al. Numerical Computation of Hyperbolic Conservation Laws Using Large Time Step Scheme with Artificial Compression Method. Arab J Sci Eng 43, 3807–3813 (2018). https://doi.org/10.1007/s13369-017-2876-5

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