Advection upstream splitting method explained

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 equations. 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 eigenvalues of the flux Jacobian matrices. The method was originally proposed by Liou and Steffen ^[1] for the typical compressible aerodynamic flows, and later substantially improved in ^{[2] [3]} to yield a more accurate and robust version. To extend its capabilities, it has been further developed in ^{[4] [5] [6]} for all speed-regimes and multiphase flow. Its variants have also been proposed.^{[7] [8]}

Features

The Advection Upstream Splitting Method has many features. The main features are:

• accurate capturing of 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 numbers.

Since the method does not specifically require eigenvectors, 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,^{[9] [10]} direct numerical simulation,^[11] multiphase flow,^[12] galactic relativistic flow^[13] 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

Notes and References

1. Liou, M.-S. and Steffen, C., “A New Flux Splitting Scheme,” J. Comput. Phys., Vol. 107, 23-39, 1993.
2. Liou, M.-S., “A Sequel to AUSM: AUSM+” J. Comput. Phys., Vol. 129, 364-382, 1996.
3. 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.
4. Liou, M.-S., “A Sequel to AUSM, Part II: AUSM+-up” J. Comput. Phys., Vol. 214, 137- 170, 2006.
5. 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.
6. 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.
7. Edwards, J. R. and Liou, M.-S., “Low-Diffusion Flux-Splitting Methods for Flows at All Speeds,” AIAA J., Vol. 36, 1610-1617, 1998.
8. 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.
9. Mary, I. and Sagaut, P., “Large Eddy Simulation of Flow Around an Airfoil Near Stall,” AIAA J., Vol. 40, 1139-1145, 2002.
10. Manoha, E., Redonnet, S., Terracol, M., and Guenanff, G., “Numerical Simulation of Aerodynamics Noise,” ECCOMAS 24–28 July 2004.
11. 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.
12. http://www.crss.ucsb.edu/ Center for Risk Studies and Safety
13. 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.