Improvement of convergence to steady state solutions of Euler equations with weighted compact nonlinear schemes

The convergence to steady state solutions of the Euler equations for weighted compact nonlinear schemes (WCNS) Deng X. and Zhang H. (2000), J. Comput. Phys. 165, 22–44 and Zhang S., Jiang S. and Shu C.-W. (2008), J. Comput. Phys. 227, 7294–7321] is studied through numerical tests. Like most other shock capturing schemes, WCNS also suffers from the problem that the residue can not settle down to machine zero for the computation of the steady state solution which contains shock waves but hangs at the truncation error level. In this paper, the techniques studied in Zhang S. and Shu. C.-W. (2007), J. Sci. Comput. 31, 273–305 and Zhang S., Jiang S and Shu. C.-W. (2011), J. Sci. Comput. 47, 216–238], to improve the convergence to steady state solutions for WENO schemes, are generalized to the WCNS. Detailed numerical studies in one and two dimensional cases are performed. Numerical tests demonstrate the effectiveness of these techniques when applied to WCNS. The residue of various order WCNS can settle down to machine zero for typical cases while the small post-shock oscillations can be removed.