Further Study on Errors in Metric Evaluation by Linear Upwind Schemes with Flux Splitting in Stationary Grids

Abstract

The importance of eliminating errors in grid-metric evaluation for high-order difference schemes has been widely recognized in recent years, and it is known from the proof by Vinokur and Yee (NASA TM 209598, 2000) that when conservative derivations of grid metric are used by Thomas, Lombard and Neier (AIAA J., 1978, 17(10) and J. Spacecraft and rocket, 1990, 27(2)), errors caused by metric evaluation could be eliminated by linear schemes when flux splitting is not considered. According to the above achievement, central schemes without the use of flux splitting could fulfill the requirement of error elimination. Difficulties will arise for upwind schemes to attain the objective when the splitting is considered. In this study, further investigations are made on three aspects: Firstly, an idea of central scheme decomposition is introduced, and the procedure to derive the central scheme is proposed to evaluate grid metrics only. Secondly, the analysis has been made on the requirement of flux splitting to acquire free-stream preservation, and a Lax-Friedrichs-type splitting scheme is proposed as an example. Discussions about current study with that by Nonomura et al. (Computers and Fluids, 2015, 107) have been made. Thirdly, for half-node- or mixed-type schemes, interpolations should be used to derive variables at half nodes. The requirement to achieve metric identity on this situation is analyzed and an idea of directionally consistent interpolation is proposed, which is manifested to be indispensable to avoid violations of metric identity and to eliminate metric-caused errors thereafter. Two numerical problems are tested, i.e., the free-stream and vortex preservation on wavy, largely randomized and triangular-like grids. Numerical results validate aforementioned theoretical outcomes.

Free-stream preserving linear-upwind and WENO schemes on curvilinear grids

“…(22) and (28), we can observe that, for the two terms in the numerical flux, since the central flux term can be applied with the symmet-rical conservative metric method, free-stream preserving can be achieved by canceling the dissipative term when the flow is uniform. In previous work, this is done either by replacing the transformed conservative variables in the difference operator of the dissipative term with the original ones and simply neglecting the effect of grid Jacobian [16,17], or freeing the metric terms at the point i + 1/2 to construct the upwind flux [15]. In this work, we split the difference operator into several local differences involving only two successive grid points.…”

“…Note that this formulation is in agreement with that of the original weighted compact nonlinear scheme (WCNS) developed by Deng and Zhang [13], in which the convectivefluxes is computed with the finite-volume version of a weighted essentially non-oscillatory (WENO) scheme [14]. Another formulation is first split the upwind scheme into a non-dissipative central part and a dissipative part, and then implementing them, respectively, with high-order finite-difference and finite-volume-like schemes by freezing Jacobian and metric terms for the entire stencil [15] or by replacing the transformed conservative variables with the original ones [16,17]. Recently, a finite-difference based free-stream preserving technique was proposed by Zhu et al [18] for WENO scheme.…”

WENO Interpolation-Based and Upwind-Biased Free-Stream Preserving Nonlinear Schemes

“…Combining the conservative metric derivation by Thomas et al [21,22], the modified WENO schemes were shown to achieve FSP [25]. In [26], a detailed investigation was carried out in order to alleviate errors in metric evaluation for arbitrary linear upwind schemes with flux splitting. In the reference, the central scheme decomposition (CSD) was presented for metric evaluations, where the requirement for flux splitting to cancel out metric-evoked errors was derived.…”

“…For example, the approach had been referred in TVD schemes [27] and high-order schemes [28][29][30][31]. In [26], some upwind linear schemes were proposed in the computations. However, they can only be applied in cases without shock waves.…”

“…Based on the analyses in [26] and the introduction of nonlinearity through WENO interpolation [32] in [17,18], new series of third-, fifth-and seventh-order WENO interpolation-based and upwind-biased schemes are proposed in this study, which are abbreviated as WENOIU for convenience. The schemes will be shown to achieve both shock capturing and FSP.…”