Limiters for Unstructured Higher-Order Accurate Solutions of the Euler Equations

Abstract

Higher-order finite-volume methods have been shown to be more efficient than secondorder methods. However no consensus has been reached on how to eliminating the oscillations caused by solution discontinuities. Essentially non-oscillatory (ENO) schemes provide a solution but are computationally expensive to implement and may not converge well for steady-state problems. This work studies the application of limiters used for second-order methods to the higher-order case. Requirements for accuracy and efficient convergence are discussed. A new limiting procedure is proposed. Results for the fourth-order accurate solution of transonic and supersonic flows demonstrate good convergence properties and significant qualitative improvement of the solution relative the second-order method. Subsonic results demonstrate the superiority of the scheme in smooth flows by the reduction in entropy production. Some aspects of the new limiter can also be successfully applied to reduce the dissipation of second-order schemes with minimal sacrifices in convergence properties.

Implicit High-Order Spectral Finite Volume Method for Inviscid Compressible Flows

“…Entropy values observed in Fig. 12 for the second-order WENO scheme are relatively close to those observed in the literature, for instance, in [36]. This reference presents entropy contours for a fully subsonic flow over a NACA 0012 airfoil, computed with second-and fourthorder spatial resolutions.…”

“…This reference presents entropy contours for a fully subsonic flow over a NACA 0012 airfoil, computed with second-and fourthorder spatial resolutions. As discussed, entropy error levels for the second-order solution in the present case are similar to those seen in [36]. For the third-order SFV scheme, however, there is no direct comparison with the cited reference, but it seems that the present results are giving entropy error levels slightly higher than the literature.…”

“…It is possible that the entropy error behavior in the present case is related to the smoothness of the grids used. The literature, considering for instance [34,36], seems to use meshes that are smoother than the one considered in the present paper, as shown in Figs. 8 and 11.…”

a posteriori stabilized sixth-order finite volume scheme for one-dimensional steady-state hyperbolic equations

“…For instance the k-exact reconstruction [3,4,34] increases the method accuracy using quadratic or cubic polynomial approximations [32,35,36]. Nevertheless, traditional TVD (Total Variation Diminishing) limiting procedures drastically reduce the order of accuracy despite the construction of alternative limiters [45,33] to enhance the quality of the solution.…”

Numerical Investigation of Flow Field Around a Ship in Manoeuvring Motion

“…Barth's limiting function for the weight factor doesn't have the differentiability characteristics, which in turn adversely affects the convergence properties of the solver. Michalak et al 21) proposed that limiting value of the weight factor to be a cubic polynomial of the limiting function, which in turn solved the problem of the lack of differentiability of Barth et als' limiter in achieving a steady state solution. This limiter provides sufficient accuracy even in smooth regions without any local extrema.…”