Smoothness-increasing accuracy-conserving (SIAC) line filtering: effective rotation for multidimensional fields

Over the past few decades there has been a strong effort towards the development of Smoothness-Increasing Accuracy-Conserving (SIAC) filters for Discontinuous Galerkin (DG) methods, designed to increase the smoothness and improve the convergence rate of the DG solution through this post-processor. The applications of these filters in multidimension have traditionally employed a tensor product kernel, allowing a natural extension of the theory developed for one-dimensional problems. In addition, the tensor product has always been done along the Cartesian axis, resulting in a filter whose support has fixed shape and orientation. This thesis has challenged these assumptions, leading to the investigation of rotated�filters: tensor product filters with variable orientation. Combining this approach with previous experiments on lower-dimension filtering, a new and computationally efficient subfamily for post-processing multidimensional data has been developed: SIAC Line filters. These filters transform the integral of the convolution into a line integral. Hence, the computational advantages are immediate: the simulation times become significantly shorter and the complexity of the algorithm design reduces to a one-dimensional problem. In the thesis, a solid theoretical background for SIAC Line �filters has been established. Theoretical error estimates have been developed, showing how Line filtering preserves the properties of traditional tensor product �filtering, including smoothness recovery and improvement in the convergence rate. Furthermore, different numerical experiments were performed, exhibiting how these filters achieve the same accuracy at significantly lower computational costs. This affords great advantages towards the applications of these filters during flow visualization; one important limiting factor of a tensor product structure is that the filter grows in support as the field dimension increases, becoming computationally expensive. SIAC Line filters have proven effi�ciency in computational performance, thus overcoming the limitations presented by the tensor product filter. The experiments carried out on streamline visualization suggest that these filters are a promising tool in scientific visualisation