Large Time Step Scheme Behaviour with Different Entropy Fix
Large Time Step Scheme Behaviour with Different Entropy Fix
Keywords:
CFL restriction, explicit scheme, inverse shock, Shock tube problem, SOD, TVD scheme, 1D Euler equationAbstract
The progress of numerical techniques for scalar and one dimensional Euler equation has been a great interest of researchers in the field of CFD for decades. In 1986, Harten developed a high resolution and efficient large time step (LTS) explicit scheme for scalar problems. Computation of nonlinear wave equation depicts that Harten’s LTS scheme is a high resolution and efficient scheme. However, computations of hyperbolic conservation laws show some spurious oscillations in the vicinities of discontinuities for larger values of CFL. Zhan Sen Qian investigated this issue and suggested to perform the inverse characteristic transformations by using the local right eigenvector matrix at each cell interface location to overcome these spurious oscillations. Harten and Qian both used Roe’s approximate Riemann solver which has less artificial viscosity than exact method at sonic points. The reduced artificial viscosity reduces the accuracy of Roe's method at sonic points. Roe's approximate Riemann solver cannot capture the finite spread of expansion fans due to the inadequate artificial viscosity at expansive sonic points. As a consequence of this expansion shocks that are nonphysical may occur. The existence of the expansion shock is said to violate the entropy condition. A variety of entropy fix formulae for Roe scheme have been addressed in the literature. In present work large time step total variation diminishing (LTS TVD) scheme developed by Harten and improved by Qian have been tested with different entropy fix and its effect has been investigated. Computed results are analyzed for merits and shortcomings of different entropy fix with large time step schemes.
References
Harten, A. The artificial compression method for computation of shocks and contact discontinuities: III. Selfadjusting 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).
Hoffmann, K., & S. Chiang. Computational Fluid Dynamics. Engineering Education System (EES), 4th ed., Vol I. Wichita, Kansas, USA (2000).
Hoffmann, K., & S. Chiang. Computational Fluid Dynamics. Engineering Education System (EES), 4thed.,Volume II. Wichita,Kansas, USA (2000).
Huang, H., C. Leey, H. Dongz, & J. Zhang. Modification and applications of a large time-step high resolution TVD scheme. In: 51st AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, AIAA, p. 2013-2077 (2013).
John, D., & J. Anderson. Computational Fluid Dynamics: The Basics with Applications. McGraw-Hill, USA (1976).
Laney, C. Computational Gasdynamics. Cambridge University Press, New York, USA, (1998).
Lax, P.D. Hyberbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves. SIAM, New York (1973).
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., & H. Mukkarum. To study large time step high resolution low dissipative schemes for hyperbolic conservation laws. Journal of Applied Fluid Mechanics, vol. 9 (2016).
Roe, P. Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics 43: 357-372 (1981).
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, 2nd ed. Taylor & Francis, CA, USA (1984).
Toro, E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd ed., Springer, Berlin (2009).
Versteeg, H., & W. Malalasekera. An Introduction to Computational Fluid Dynamics: The Finite Volume Method, 2nd ed. Pearson Prentice Hall, England, United Kingdom (2007).
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, CA, 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).
ZhanSen, Q.C.L. A class of large time step godunov scheme for hyperbolic conservation laws and applications.Journal of Computational Physics 230: 7418-7440 (2011).
ZhanSen, Q.C.-H.L. On large time step tvd scheme for hyperbolic conservation laws and its efficiencyevaluation.Journal of Computational Physics 231: 7415-7430 (2012).
