# Computation of LAX Shock Tube Test Case through Large Time Step Scheme with Compressive Limiters

## LTS Scheme with Compressive Limiters

## Keywords:

TVD Scheme, Numerical Dissipation, Shock Tube Problem, LAX## Abstract

Computation of accurate and efficient numerical results for space vehicle design and analysis is a challenging task because it takes large computational time to predict complex flow physics of space vehicle. Space vehicle travels through continuum as well as rarefied region during flight. Continuum region aerodynamics can be predicted by solving Navier Stokes equation. Explicit schemes require low computational hardware facility but increase computational time by limiting time step to a certain limit defined by stability criteria. An extensive research is being done for last three decades to overcome this stability restriction. Initially, Harten proposed a large time step Total Variation Diminishing (TVD) second order accurate (2K+3) point scheme with explicit formulation under a CFL restriction of K. Harten’s developed large time step scheme and its modified Qian’s form have been tested with minmod limiter extensively. However detailed analysis of these schemes with more compressive limiter is still in

progress. Present research investigates Qian’s modified large time step scheme behavior with compressive limiters for complex flow physics. Shock tube problem with Lax boundary condition is computed to point out advantages and short comings of Qian’s proposed modified scheme.

## References

John, D., J. Anderson. Computational Fluid Dynamics: The Basics with Applications. McGraw- Hill, University of Maryland ,USA (1995).

Lax, P. D. Hyberbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. New York: SIAM (1973).

Laney, C. Computational Gasdynamics. Cambridge University Press, New York, USA, (1998).

Hoffmann, K., & Chiang, S. Computational Fluid Dynamics. Engineering Education System(EES), Fourth Edition,Volume I. Wichita,Kansas,USA (2000).

Hoffmann, K., & S. Chiang, Computational Fluid Dynamics. Engineering Education System(EES), Fourth Edition,Volume II. Wichita,Kansas,USA (2000).

Toro, P. E. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Third Edition, University of Trento, Italy, Springer Dordrecht Heidelberg London New York. (2009).

Harten, A. The Artificial Compression Method for Computation of Shocks and Contact Discontinuities: III. Self-Adjusting Hybrid Schemes, Mathematics of Computation,32 (142): 363-389 (1978).

Harten, A. High Resolution Schemes for Hyperbolic Conservation Laws, Journal of Computational Physics, 135: 260-278 (1997).

Harten, A. On a Large Time-Step High Resolution Scheme. Mathematics of Computation, 46(174):379-399 (1986).

Roe, P. Approximate Riemann Solvers, Parameter Vectors,and Difference Schemes, Journal of Computational Physics 43: 357-372 (1981).

Yee, H. Upwind and Symmetric Shock-Capturing Schemes, NASA Technical Memorandum, 130: 89464 (1987).

Yee, H.C, A Class of High-Resolution Explicit and Implicit Shock-Capturing Methods, Ames Research Center, NASA Technical Memorandum 101088 , 228., Moffett Field, California, USA (February 1989).

Yee, H., G. Klopfer, & J. Montagn, High-Resolution Shock-Capturing Schemes for lnviscid and Viscous Hypersonic Flows, NASA Technical Memorandum 100097, 38. USA (April 1988).

Yee, H., N. Sandham, & M. Djomehri, Low Dissipative High Order Shock Capturing Methods using Characteristic Based Filter, Journal of Computational Physics, 150: 199-238 (1999).

SOD, G.A. A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws, Journal of Computational Physics, 27: (1978).

Sweby, P. K. High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws. Journal on Numerical Analysis, SIAM, 21(5): 995- 1011 (1984).

Tannehill, J., D. Anderson, & H. Pletcher, Computational Fluid Mechanics and Heat Transfer, Second Edition, Taylor & Francis, USA (1984).

ZhanSen Qian, C.L, A class of Large Time Step Godunov Scheme for Hyperbolic Conservation Laws and Applications, Journal of Computational Physics, 230: 7418-7440 (2011).

Qian, Z, On Large Time Step TVD Scheme for Hyperbolic Conservation Laws and its Efficiency Evaluation. Journal of Computational Physics, 231:7415-7430 (2012).

Huang, H., C. Leey., H. Dongz, & J. Zhang, Modification and Applications of a Large Time-Step High Resolution TVD Scheme, 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA:2013-0077 (2013).

Versteeg, H., & W. Malalasekera, An Introduction to Computational Fluid Dynamics: The Finite Volume Method, Second Edition, PEARSON Prentice Hall, England, United Kingdom (2007).

Mukkarum, H., U.H. Ihtram, & F. Noor, Efficient and Accurate Scheme for Hyperbolic Conservation Laws, International Journal of Mathematical Models and Methods in Applied Sciences, NUAN, 9: 504-511 (2015).

Noor, F., & Mukkarum, H. To Study Large Time Step High Resolution Low Dissipative Schemes for Hyperbolic Conservation Laws. Journal of Applied Fluid Mechanics, vol. 9: 2073-2081 (2016).

Wakif, A., Z. Boulahia, & R. Sehaqui, A semianalytical analysis of electro-thermo-hydrodynamic stability in dielectric nanofluids using Buongiorno’s mathematical model together with more realistic boundary conditions, Results in Physics, 9: 1438–1454 (2018).

Wakif, A., Z. Boulahia., A. Ali., R.A. Eid, & R. Sehaqui, Numerical Analysis of the Unsteady Natural Convection MHD Couette Nanofluid Flow in the Presence of Thermal Radiation Using Single and Two-Phase Nanofluid Models for Cu–Water Nanofluids, International Journal of Applied and Computational Mathematics, 4(3): (2018).

Abro, K.A., A.A. Memon,S.H. Abro., I. Khan, & I.Tlili, I, Enhancement of heat transfer rate of solar energy via rotating Jeffrey nanofluids using Caputo–Fabrizio fractional operator: An application to solar energy. Energy Reports, 5: 41–49(2019).

Abro, K.A., & A. Yıldırım, Fractional Treatment of Vibration Equation Through Modern Analogy of Fractional Differentiations Using Integral Transforms. Iranian Journal of Science and Technology, Transactions A: Science. (2019) doi:10.1007/s40995-019-00687-4

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*Proceedings of the Pakistan Academy of Sciences: A. Physical and Computational Sciences*,

*56*(1), 29–37. Retrieved from https://ppaspk.org/index.php/PPAS-A/article/view/149