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The '''Advection Upstream Splitting Method (AUSM)''' is a [[numerical method]] used to solve the [[advection]] equation in [[computational fluid dynamics]]. It is particularly useful for simulating compressible flows with shocks and discontinuities.
The AUSM is developed as a numerical [[inviscid]] [[flux]] function for solving a general system of [[conservation equation]]s. It is based on the upwind concept and was motivated to provide an alternative approach to other upwind methods, such as the [[Godunov method]], flux difference splitting methods by Roe, and Solomon and Osher, flux vector splitting methods by Van Leer, and Steger and Warming.
The AUSM first recognizes that the inviscid flux consist of two physically distinct parts, i.e., convective and pressure fluxes. The former is associated with the flow ([[advection]]) speed, while the latter with the acoustic speed; or respectively classified as the linear and nonlinear fields. Currently, the convective and pressure fluxes are formulated using the [[eigenvalue]]s of the flux [[Jacobian matrix|Jacobian matrices]]. The method was originally proposed by Liou and Steffen <ref>Liou, M.-S. and Steffen, C., “A New Flux Splitting Scheme,” J. Comput. Phys., Vol. 107, 23-39, 1993.</ref> for the typical compressible aerodynamic flows, and later substantially improved in <ref name="firstName">Liou, M.-S., “A Sequel to AUSM: AUSM+” J. Comput. Phys., Vol. 129, 364-382, 1996.</ref><ref>Wada, Y. and Liou, M.-S., “An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities,” SIAM J. Scientific Computing, Vol. 18, 633-657, 1997.</ref> to yield a more accurate and robust version. To extend its capabilities, it has been further developed in <ref name="secondName">Liou, M.-S., “A Sequel to AUSM, Part II: AUSM+-up” J. Comput. Phys., Vol. 214, 137- 170, 2006.
</ref><ref>Edwards, J. R., Franklin, R., and Liou, M.-S., “Low-Diffusion Flux-Splitting Methods for Real Fluid Flows with Phase Transitions,” AIAA J., Vol. 38, 1624-1633, 2000.</ref><ref>Chang, C.-H. and Liou, M.-S., “A New Approach to the Simulation of Compressible Multifluid Flows with AUSM+ Scheme,” AIAA Paper 2003-4107, 16th AIAA CFD Conference, Orlando, FL, June 23–26, 2003.</ref> for all speed-regimes and [[multiphase flow]]. Its variants have also been proposed.<ref>Edwards, J. R. and Liou, M.-S., “Low-Diffusion Flux-Splitting Methods for Flows at All Speeds,” AIAA J., Vol. 36, 1610-1617, 1998.</ref><ref>Kim, K. H., Kim, C., and Rho, O., “Methods for the Accurate Computations of Hypersonic Flows I. AUSMPW+ Scheme,” J. Comput. Phys., Vol. 174, 38-80, 2001.</ref>
==Features==
The Advection Upstream Splitting Method has many features. The main features are:
*accurate capturing of [[Shocks and discontinuities (magnetohydrodynamics)|shock and contact discontinuities]]
*[[entropy]]-satisfying solution
*positivity-preserving solution
*algorithmic simplicity (not requiring explicit eigen-structure of the flux Jacobian matrices) and straightforward extension to additional conservation laws
*free of “carbuncle” phenomena
*uniform accuracy and convergence rate for all [[Mach number]]s.
Since the method does not specifically require [[eigenvector]]s, it is especially attractive for the system whose eigen-structure is not known explicitly, as the case of two-fluid equations for multiphase flow.
==Applications==
The AUSM has been employed to solve a wide range of problems, low-Mach to [[hypersonic]] [[aerodynamics]], [[large eddy simulation]] and [[aero-acoustics]],<ref>Mary, I. and Sagaut, P., “Large Eddy Simulation of Flow Around an Airfoil Near Stall,” AIAA J., Vol. 40, 1139-1145, 2002.</ref><ref>Manoha, E., Redonnet, S., Terracol, M., and Guenanff, G., “Numerical Simulation of Aerodynamics Noise,” ECCOMAS 24–28 July 2004.</ref> direct numerical simulation,<ref>Billet, G. and Louedin, O., “Adaptive Limiters for Improving the Accuracy of the MUSCL Approach for Unsteady Flows,” J. Comput. Phys., Vol. 170, 161-183, 2001.</ref> multiphase flow,<ref>[http://www.crss.ucsb.edu/ Center for Risk Studies and Safety] {{webarchive |url=https://web.archive.org/web/20060424060610/http://www.crss.ucsb.edu/ |date=April 24, 2006 }}, University of California (Santa Barbara)</ref> galactic relativistic flow<ref>Wada, K. and Koda, J., “Instabilities of Spiral Shock – I. Onset of Wiggle Instability and its Mechanism,” [[Monthly Notices of the Royal Astronomical Society]], Vol. 349, 270-280 (11), 2004.</ref> etc.
==See also==
*[[Euler equations]]
*[[Finite volume method]]
*[[Flux limiter]]
*[[Godunov's theorem]]
*[[High resolution scheme]]
*[[Numerical method of lines]]
*[[Sergei K. Godunov]]
*[[Total variation diminishing]]
==References==
{{Reflist}}
{{DEFAULTSORT:Advection upstream splitting method}}
[[Category:Computational fluid dynamics]]
[[Category:Numerical differential equations]]