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Advection upstream splitting method
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<h2 id="features">Features</h2> <p>The Advection Upstream Splitting Method has many features. The main features are: </p> <ul><li>accurate capturing of <a href="/facts/Shocks_and_discontinuities_(magnetohydrodynamics)/8ul1Plh5">shock and contact discontinuities</a></li> <li><a href="/facts/Entropy/s52IXeFz">entropy</a>-satisfying solution</li> <li>positivity-preserving solution</li> <li>algorithmic simplicity (not requiring explicit eigen-structure of the flux Jacobian matrices) and straightforward extension to additional conservation laws</li> <li>free of “carbuncle” phenomena</li> <li>uniform accuracy and convergence rate for all <a href="/facts/Mach_number/Ee01MIgw">Mach numbers</a>.</li></ul> <p>Since the method does not specifically require <a href="/facts/Eigenvector/8TjEoT8u">eigenvectors</a>, it is especially attractive for the system whose eigen-structure is not known explicitly, as the case of two-fluid equations for multiphase flow. </p> <h2 id="applications">Applications</h2> <p>The AUSM has been employed to solve a wide range of problems, low-Mach to <a href="/facts/Hypersonic/QUsQbhur">hypersonic</a> <a href="/facts/Aerodynamics/Cn6Lnl3H">aerodynamics</a>, <a href="/facts/Large_eddy_simulation/ULBNg1SY">large eddy simulation</a> and <a href="/facts/Aero-acoustics/FJN943nB">aero-acoustics</a>,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> direct numerical simulation,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> multiphase flow,<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> galactic relativistic flow<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> etc. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Euler_equations/F1zh068s">Euler equations</a></li> <li><a href="/facts/Finite_volume_method/6JfQMucX">Finite volume method</a></li> <li><a href="/facts/Flux_limiter/vWBzcJx0">Flux limiter</a></li> <li><a href="/facts/Godunov%27s_theorem/CoLFQRm7">Godunov's theorem</a></li> <li><a href="/facts/High_resolution_scheme/N56RRHnS">High resolution scheme</a></li> <li><a href="/facts/Numerical_method_of_lines/FrKcwfYV">Numerical method of lines</a></li> <li><a href="/facts/Sergei_K._Godunov/IaYktIaI">Sergei K. Godunov</a></li> <li><a href="/facts/Total_variation_diminishing/BxKKPAkR">Total variation diminishing</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Liou, M.-S. and Steffen, C., “A New Flux Splitting Scheme,” J. Comput. Phys., Vol. 107, 23-39, 1993. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Liou, M.-S., “A Sequel to AUSM: AUSM+” J. Comput. Phys., Vol. 129, 364-382, 1996. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Liou, M.-S., “A Sequel to AUSM, Part II: AUSM+-up” J. Comput. Phys., Vol. 214, 137- 170, 2006. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>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. <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Edwards, J. R. and Liou, M.-S., “Low-Diffusion Flux-Splitting Methods for Flows at All Speeds,” AIAA J., Vol. 36, 1610-1617, 1998. <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>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. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Mary, I. and Sagaut, P., “Large Eddy Simulation of Flow Around an Airfoil Near Stall,” AIAA J., Vol. 40, 1139-1145, 2002. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Manoha, E., Redonnet, S., Terracol, M., and Guenanff, G., “Numerical Simulation of Aerodynamics Noise,” ECCOMAS 24–28 July 2004. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>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. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Center for Risk Studies and Safety Archived April 24, 2006, at the Wayback Machine, University of California (Santa Barbara) <a href="http://www.crss.ucsb.edu/" target="_blank">http://www.crss.ucsb.edu/</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>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. <a href="/wiki/Monthly_Notices_of_the_Royal_Astronomical_Society" target="_blank">/wiki/Monthly_Notices_of_the_Royal_Astronomical_Society</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> </ol>