Improvement of Convergence to Steady State Solutions of Euler Equations with the WENO Schemes

The convergence to steady state solutions of the Euler equations for high order weighted essentially non-oscillatory (WENO) finite difference schemes with the Lax-Friedrichs flux splitting [Jiang G.-S. and Shu C.-W. (1996), J. Comput. Phys. 126, 202-228] is investigated. Numerical evidence in [Zhang S. and Shu C.-W. (2007), J. Sci. Comput. 31, 273-305] indicates that there exist slight post-shock oscillations when we use high order WENO schemes to solve problems containing shock waves. Even though these oscillations are small in their magnitude and do not affect the “essentially non-oscillatory” property of the WENO schemes, they are indeed responsible for the numerical residue to hang at the truncation error level of the scheme instead of settling down to machine zero. Different from the strategy adopted in [Zhang S. and Shu C.-W. (2007), J. Sci. Comput. 31, 273-305], in which a new smoothness indicator was introduced to facilitate convergence to steady states, in this paper we study the effect of the local characteristic decomposition on steady state convergence. Numerical tests indicate that the slight post shock oscillation has a close relationship with the local characteristic decomposition process. When this process is based on an average Jacobian at the cell interface using the Roe average, as is the standard procedure for WENO schemes, such post shock oscillation appears. If we instead use upwind biased interpolation to approximate the physical variables including the velocity and enthalpy on the cell interface to compute the left and right eigenvectors of the Jacobian for the local characteristic decomposition, the slight post shock oscillation can be removed or reduced significantly and the numerical residue settles down to lower values than other WENO schemes and can reach machine zero for many test cases. This new procedure is also effective for higher order WENO schemes and for WENO schemes with different smoothness indicators.