High-Order Behaviors of Weighted Compact Fifth-Order Nonlinear Schemes

Abstract

T HE typical theory for the finite difference method is that partial difference equations (PDEs) are discretized in space using a Taylor series expansion and solving it for the variables at discrete points. The derivatives are written as functions of variables on neighboring points. The approximations are said to be high-order accurate when the power p of the leading truncate error O h is greater than two. Because of the advantages of high-flow structure resolution and nice discontinuity capturing precision, high-order accurate schemes are playing important roles in computational fluid dynamics (CFD). Nonlinear high-order schemes for computing turbulence and aeroacoustic flows, which contain shock, have received more and more attention since the last decade. Although compact schemes have been derived by some researchers like Lele [1] and Gaitonde and Shang [2], theywere poorly used in capturing the shock owing to their linear property in construction. Cockburn and Shu [3] proposed fourth-order compact nonlinear schemes based on a total variation diminishing (TVD) and total variation bounded (TVB) concept. Hu et al. [4] developed a fifth-order upwind compact scheme to solve the acoustic phenomena arising from shock-vortex interactions. According to the principles from the physical consideration, Zhang et al. [5] constructed a mixing method, which degenerated into the second-order nonoscillatory, containing no free parameters and dissipative (NND) scheme in the shock region and retained highorder accuracy elsewhere. An appropriate weight technique treated in high-order scheme construction may also differentiate the vortex in the smooth region and shocks robustly. Levy et al. [6] applied fourth-order central weighted essentially nonoscillatory (WENO) to hypersonic flows. Shu [7] used fifth-order WENO for shock problems. Deng [8] and other researchers have developed a variety of compact high-order accurate schemes. A type of one-parameter linear dissipative compact scheme (DCS)was derived as to damp out the dispersive and parasite errors in the high-wave-number regions when central compact schemes were used. Although the linear compact schemes often caused numerical oscillations in the vicinity of discontinuities, a fourth-order cell-centered compact nonlinear finite difference scheme (CNS) [9] was then derived. An adaptive interpolation of variables at cell edges was designed, which automatically jumped to local one as discontinuities were encountered. It was a way to make the overall compact scheme capture discontinuities in a nonoscillatory manner. Afterward, using the weighted technique of Jiang and Shu [10] for interpolation at cell edge, fourthand fifth-order weighted compact nonlinear schemes (WCNS) were developed [11,12]. They were more efficient than CNS, for two tridiagonal inversions were cancelled and no logical algorithms were needed. To eliminate the unphysical oscillations in the vortex field near the shock wave when inviscid flow was computed, Deng constructed high-order dissipative weighted compact nonlinear schemes (DWCNS) [13], which improved the ability of restraining nonphysical oscillation in the vorticity field. For viscous flow computations, as proposed in [13], WCNS-5 and WCNS-E-5 would better be used. This Note shows the high-order features of WCNS including WCNS-E-5 and WCNS-5 with such typical high-order schemes as the explicit upwind biased fifth-order scheme (EUW-5) [14], Pade [1], and other special schemes like WENOs [6,7], and the explicit fifth-order shock-fitting upwind scheme [15]. We do the Fourier analysis to discussWCNS-E-5 andWCNS-5 with EUW-5 and Pade. Owing to nontridiagonal inversion,WCNS-E-5 acts more efficiently than WCNS-5 during computations. Therefore, in Sec. IV, we use WCNS-E-5 to solve the multidimensional inviscid/viscous flows. WCNS-E-5 obtains several numerical results to show the good performances with those by WENOs and the explicit fifth-order shock-fitting upwind scheme.