MULTI-DIMENSIONAL FILTERING: REDUCING THE DIMENSION THROUGH ROTATION

Over the past few decades there has been a strong effort toward the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for discontinuous GalerkM (DC) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this postprocessor. These advantages can be exploited during flow visualization, far example, by applying the SIAC filter to DC data before streamline computations [M. Steffen, S. Curtis, R. M. Kirby, and J. K. Ryan, IEEE Trans. Vis. Cornput. Graphics, 14 (2008), pp. 680-692]. However, introducing these filters in engineering applications can be challenging since a tensor product filter grows in support size as the field dimension increases, becoming computationally expensive. As an alternative, [D. Walfisch, J. K. Ryan, R. M. Kirby, and R. Haimes, J. Sci. Comput., 38 (2009), pp. 164 184] proposed a univariate filter implemented along the streamline curves. Until now, this technique; remained a numerical experiment. In this paper we introduce the line SIAC filter and explore how the orientation, structure, and filter size affect the order of accuracy and global errors. We present theoretical error estimates showing how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs, becoming an attractive tool for the scientific visualization comnmnity.