One-sided Post-processing for the Discontinuous Galerkin Method Using ENO Type Stencil Choosing and the Local Edge Detection Method

Abstract

], a one-sided post-processing technique for the discontinuous Galerkin method was introduced for reconstructing solutions near computational boundaries and discontinuities in the boundaries, as well as for changes in mesh size. This technique requires prior knowledge of the discontinuity location in order to determine whether to use centered, partially one-sided, or one-sided post-processing. We now present two alternative stencil choosing schemes to automate the choice of post-processing stencil. The first is an ENO type stencil choosing procedure, which is designed to choose centered post-processing in smooth regions and one-sided or partially one-sided post-processing near a discontinuity, and the second method is based on the edge detection method designed by Archibald, Gelb, and Yoon [Archibald, R., Gelb, A., and Yoon, J. (2005). SIAM J. Numeric. Anal. 43, 259-279; Archibald, R., Gelb, A., and Yoon, J. (2006). Appl. Numeric. Math. (submitted)]. We compare these stencil choosing techniques and analyze their respective strengths and weaknesses. Finally, the automated stencil choices are applied in conjunction with the appropriate post-processing procedures and it is determine that the resulting numerical solutions are of the correct order.

Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering

“…The SIAC filtering increases the inter-element continuity up to C k−1 and raises the convergence rate of the DG solution from order k + 1 to order 2k+1 (i.e., superconvergence) for linear [9] and nonlinear [15,18,17] hyperbolic equations solved over a uniform mesh, and variable coefficient euqations [17,24]. The proof has also been extended for local derivative computation [29] and boundary filtering [30,4,13]. Convergence properties of SIAC filtering and its effectiveness have been widely studied in the literature [9,19,6,12,14,16].…”

Edge Detection by Adaptive Splitting II. The Three-Dimensional Case

“…[8,10,33]. Other application areas include the determination of the interface which separates two or more different material media [12,30] and the enhancement of some numerical algorithms [2].…”

“…We need some previous definitions e, = (x! -25) 2 Figure 4 shows the interface (F J ) of / 2 . The results obtained are reported in Table 5.…”

Determining the locations and discontinuities in the derivatives of functions

“…It has been shown in [10,18,20] that spectral accuracy can be recovered in smooth regions for nonlinear conservation laws using the spectral viscosity method. It has also been recently demonstrated that high order accuracy in smooth regions can be recovered with suitable post-processing of the WENO method [11], and of the discontinuous Galerkin method [1]. Hence we are motivated to develop a method that determines not only the discontinuities in piecewise smooth functions, but the discontinuities in each successive derivative function as well.…”