A Very Efficient Class of HOC Schemes for the One-Dimensional Euler Equations of Gas Dynamics

Abstract:

This manuscript introduces a class of higher order compact schemes for the solution of one dimensional (1-D) Euler equations of gas dynamics. These schemes are fourth order accurate in space and second or lower order accurate in time, depending on a weighted average parameter μ. The robustness and efficiency of our proposed schemes have been validated by applying them to three different shock-tube problems of gas dynamics, including the famous SOD shock-tube problem. Later on, the 1-D convergent-divergent nozzle problem (De laval nozzle problem) is also considered and numerical simulations are performed. In all the cases, our computed numerical solutions are found to be in excellent match with the exact solutions or available results in the existing literature. Overall the schemes are found to be efficient and accurate.

• Periodical: Applied Mechanics and Materials (Volumes 813-814)
• Pages: 643-651
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.813-814.643
• Online since: November 2015
• Authors: Bidyut B. Gogoi
• Keywords: Euler Equations, Gas Dynamics, Higher Order Compact, Nozzle, Shock-Tube, Stability

References

[1] S. Abarbanel, A. Kumar, Compact High-Order Schemes for the Euler Equations, J. Sci. Comp., 3 (1988) 275-288.

DOI: 10.1007/bf01061287

[2] M.A.H. Mohamad, S. Basri, B. Basuno, One-dimensional high order compact method for solving Euler's equations, Proc. of the 4th International Meeting of Advances in Thermofluids, Melaka, Malaysia, 2011, pp.950-962.

DOI: 10.1063/1.4704291

[3] C.W. Shu, High-order Finite Difference and Finite Volume WENO Schemes and Discontinuous Galerkin Methods for CFD, Int. J. Comp. Fluid Dyn., 17 (2003) 107-118.

DOI: 10.1080/1061856031000104851

[4] A. Rezgui, P. Cinnella, A. Lerat, Third-order accurate finite volume schemes for Euler computations on curvilinear meshes, Comp. & Fluids, 30 (2001) 875-901.

DOI: 10.1016/s0045-7930(01)00033-0

[5] C.O. Gooch, A. Nejat, K. Michalak, Obtaining and Verifying High-Order Unstructured Finite Volume Solutions to the Euler Equations, AIAA J., 47 (2009) 2105-2120.

DOI: 10.2514/1.40585

[6] F. Bassi, S. Rebay, High-order accurate discontinuous finite element solution of the 2D Euler equations. J. Comp. Phys., 138 (1997) 251-285.

DOI: 10.1006/jcph.1997.5454

[7] L. Wang, D.J. Mavriplis, Implicit solution of the unsteady Euler equations for high-order accurate discontinuous Galerkin discretizations, J. Comp. Phys., 225 (2007) 1994-(2015).

DOI: 10.1016/j.jcp.2007.03.002

[8] N. Kwatra, S. Jonathan, J.T. Gretarsson, R. Fedkiw, A method for avoiding the acoustic time step restriction in compressible flow, J. Comp. Phys., 228 (2009) 4146-4161.

DOI: 10.1016/j.jcp.2009.02.027

[9] A.M. Elfaghi, W. Asrar, A.A. Omar, Higher order compact flowfield dependent variation solution of one-dimensional problems, Eng. App. Comp. Fluid Mech., 4 (2010) 434-440.

DOI: 10.1080/19942060.2010.11015330

[10] G. Stipcich, C. Liu, High-order mixed weighted compact and non compact scheme for shock and small length scale interaction, Int. J. Comp. Maths., 90 (2013) 376-407.

DOI: 10.1080/00207160.2012.718336

[11] J.D. Anderson Jr., Computational Fluid Dynamics, McGraw-Hill, Inc., New York, (1995).

[12] A. Xu, Numerical solutions of quasi-one-dimensional nozzle flows, Proc. of the 23rd International Congress of Theoretical and Applied Mechanics (ICTAM - 2012), Beijing, China, 2012, pp.1-10.