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

In a previous paper by Ryan and Shu [Ryan, J. K., and Shu, C.-W. (2003). Methods Appl. Anal. 10(2), 295–307], 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.