Quantification of Errors Introduced in the Numerical Approximation and Implementation of Smoothness-Increasing Accuracy Conserving (SIAC) Filtering of Discontinuous Galerkin (DG) Fields

The discontinuous Galerkin (DG) method continues to maintain heightened levels of interest within the simulation community because of the discretization flexibility it provides. Although one of the fundamental properties of the DG methodology and arguably its most powerful property is the ability to combine high-order discretizations on an interelement level while allowing discontinuities between elements, this flexibility generates a plethora of difficulties when one attempts to post-process DG fields for analysis and evaluation of scientific results. Smoothness-increasing accuracy-conserving (SIAC) filtering enhances the smoothness of the field by eliminating the discontinuity between elements in a way that is consistent with the DG methodology; in particular, high-order accuracy is preserved and in many cases increased. Fundamental to the post-processing approach is the convolution of a spline-based kernel against a DG field. This paper presents a study of the impact of numerical quadrature approximations on the resulting convolution. We discuss both theoretical estimates as well as empirical results which demonstrate the efficacy of the post-processing approach when different levels and types of quadrature approximation are used. Finally, we provide some guidelines for effective use of SIAC filtering of DG fields when used as input to common post-processing and visualization techniques. This paper is in honor of our mentor and collaborator Professor David Gottlieb, who taught us that implementations require approximations, approximations lead to “numerical crimes”, and as with all crimes, numerical crimes come at a cost. H. Mirzaee · R.M. Kirby ( ) School of Computing, University of Utah, Salt Lake City, UT, USA e-mail: [email protected] H. Mirzaee e-mail: [email protected] J.K. Ryan Delft Institute of Applied Mathematics, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands e-mail: [email protected] 448 J Sci Comput (2010) 45: 447–470