Skip to main content Accessibility help

AUSM-Based High-Order Solution for Euler Equations

  • Angelo L. Scandaliato (a1) and Meng-Sing Liou (a2)


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.


Corresponding author



Hide All
[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.
[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.
[3]Gerolymos, G.A., Senechal, D., and Vallet, I., Very-high-order WENO schemes. AIAA Aerospace Science Meeting, 47 (2009).
[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.
[5]Harten, A., High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49 (1983), 357–393.
[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.
[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.
[8]Jiang, G.‐S. and ‐W Shu, C., Efficient implementation of weighted ENO schemes. J. Comput. Phys., 126 (1996), 202–228.
[9]Liou, M.‐S., A sequel to AUSM, part II: AUSM₊₋up for all speeds. J. Comput. Phys., 214 (2006), 137–170.
[10]Liou, M.‐S., A sequel to AUSM: AUSM₊. J. Comput. Phys., 129 (1996), 364–382.
[11]Liou, M.‐S., Open problems in numerical fluxes: Proposed resolutions. In 20th AIAA Computational Fluid Dynamics Conference (2011).
[12]Liou, M.‐S., Progress towards an improved CFD method: AUSM₊. AIAA paper 1995-1701-CP, in 12th AIAA Computational Fluid Dynamics Conference (1995).
[13]Liou, M.‐S. and Steffen Jr., C.J., A new flux splitting scheme. J. Comput. Phys., 107 (1993), 23–39.
[14]Roe, P.L., Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys., 43 (1981), 357–372.
[15]Shu, C.–W. and Osher, S., Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys., 77 (1988), 439–471.
[16]Suresh, A. and Huynh, H.T., Accurate monotonicity-preserving schemes with Runge-Kutta time stepping. J. Comput. Phys., 136 (1997), 83–99.
[17]Titarev, V.A. and Toro, E.F., Finite-volume WENO schemes for 3-D conservation laws. J. Comput. Phys., 201 (2004), 238–260.
[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 712 2002).
[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.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed