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AUSM-Based High-Order Solution for Euler Equations

Published online by Cambridge University Press:  20 August 2015

Angelo L. Scandaliato  and
Meng-Sing Liou
Angelo L. Scandaliato*
Affiliation:
Ohio Aerospace Institute, Cleveland, OH, 44142, USA
Meng-Sing Liou*
Affiliation:
NASA Glenn Research Center, Cleveland, OH, 44135, USA
*
Email:ascandal@ucsd.edu
Corresponding author.Email:meng-sing.liou@nasa.gov
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Abstract

In this paper we demonstrate the accuracy and robustness of combining the advection upwind splitting method (AUSM), specifically AUSM+-UP, with high-order upwind-biased interpolation procedures, the weighted essentially non-oscillatory (WENO-JS) scheme and its variations, and the monotonicity preserving (MP) scheme, for solving the Euler equations. MP is found to be more effective than the three WENO variations studied. AUSM+-UP is also shown to be free of the so-called “carbuncle” phenomenon with the high-order interpolation. The characteristic variables are preferred for interpolation after comparing the results using primitive and conservative variables, even though they require additional matrix-vector operations. Results using the Roe flux with an entropy fix and the Lax-Friedrichs approximate Riemann solvers are also included for comparison. In addition, four reflective boundary condition implementations are compared for their effects on residual convergence and solution accuracy. Finally, a measure for quantifying the efficiency of obtaining high order solutions is proposed; the measure reveals that a maximum return is reached after which no improvement in accuracy is possible for a given grid size.

Keywords

6576Shock capturingadvection upwind splittingEuler equationsweighted essentially non-oscillatorymonotonicity preserving
Type
Research Article
Information
Communications in Computational Physics , Volume 12 , Issue 4 , October 2012 , pp. 1096 - 1120
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Balsara, D.S. and Shu, C.-W., Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys., 160 (2000), 405–452.Google Scholar
[2]Borges, R., Carmona, M., Costa, B., and Don, W.S., An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys., 227 (2008), 3191–3211.Google Scholar
[3]Gerolymos, G.A., Senechal, D., and Vallet, I., Very-high-order WENO schemes. AIAA Aerospace Science Meeting, 47 (2009).Google Scholar
[4]Godunov, S.K., A finite difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb., 47 (1959), 357–393.Google Scholar
[5]Harten, A., High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49 (1983), 357–393.Google Scholar
[6]Henderson, L.F., Vasilev, E.I., Ben-Dor, G., and Elperin, T., The wall-jetting effect in mach reflection: Theoretical consideration and numerical investigation.J. Fluid Mech., 479 (2003), 259–286.Google Scholar
[7]Henrick, A.K., Aslam, T.D., and Powers, J.M., Mapped weighted essentially non-oscillatory schemes: Achieving optimal order near critical points. J. Comput. Phys., 207 (2005), 542–567.Google Scholar
[8]Jiang, G.‐S. and ‐W Shu, C., Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126 (1996), 202–228.Google Scholar
[9]Liou, M.‐S., A sequel to AUSM, part II: AUSM₊₋up for all speeds. J. Comput. Phys., 214 (2006), 137–170.Google Scholar
[10]Liou, M.‐S., A sequel to AUSM: AUSM₊. J. Comput. Phys., 129 (1996), 364–382.Google Scholar
[11]Liou, M.‐S., Open problems in numerical fluxes: Proposed resolutions. In 20th AIAA Computational Fluid Dynamics Conference (2011).Google Scholar
[12]Liou, M.‐S., Progress towards an improved CFD method: AUSM₊. AIAA paper 1995-1701-CP, in 12th AIAA Computational Fluid Dynamics Conference (1995).Google Scholar
[13]Liou, M.‐S. and Steffen Jr., C.J., A new flux splitting scheme. J. Comput. Phys., 107 (1993), 23–39.Google Scholar
[14]Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43 (1981), 357–372.Google Scholar
[15]Shu, C.–W. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys., 77 (1988), 439–471.CrossRefGoogle Scholar
[16]Suresh, A. and Huynh, H.T., Accurate monotonicity-preserving schemes with Runge-Kutta time stepping. J. Comput. Phys., 136 (1997), 83–99.Google Scholar
[17]Titarev, V.A. and Toro, E.F., Finite-volume WENO schemes for 3-D conservation laws. J. Comput. Phys., 201 (2004), 238–260.Google Scholar
[18]van der Vegt, J.J.W and van der Ven, H., Slip flow boundary conditions in discontinuous Galerkin discretizations of the Euler equations of gas dynamics. World Congress on Comput. Mech., 5 (July 7–12 2002).Google Scholar
[19]Zhang, S. and Shu, C.‐W., A new smoothness indicator for the WENO schemes and its effect on the convergence to steady state solutions. J. Sci. Comput., 31 (2007), 273–305.Google Scholar