Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Discontinuous Galerkin Solutions: Improved Errors Versus Higher-Order Accuracy
James King
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Hanieh Mirzaee
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Jennifer K. Ryan
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Robert M. Kirby
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J.K. Ryan ( ) Delft Institute of Applied Mathematics, Delft University of Technology
, Mekelweg 4, 2628 CD Delft,
The Netherlands
Smoothness-increasing accuracy-conserving (SIAC) filtering has demonstrated its effectiveness in raising the convergence rate of discontinuous Galerkin solutions from order k + 21 to order 2k + 1 for specific types of translation invariant meshes (Cockburn et al. in Math. Comput. 72:577-606, 2003; Curtis et al. in SIAM J. Sci. Comput. 30(1):272289, 2007; Mirzaee et al. in SIAM J. Numer. Anal. 49:1899-1920, 2011). Additionally, it improves the weak continuity in the discontinuous Galerkin method to k 1 continuity. Typically this improvement has a positive impact on the error quantity in the sense that it also reduces the absolute errors. However, not enough emphasis has been placed on the difference between superconvergent accuracy and improved errors. This distinction is particularly important when it comes to understanding the interplay introduced through meshing, between geometry and filtering. The underlying mesh over which the DG solution is built is important because the tool used in SIAC filteringconvolutionis scaled by the geometric mesh size. This heavily contributes to the effectiveness of the post-processor. In this paper, we present a study of this mesh scaling and how it factors into the theoretical errors. To accomplish the large volume of post-processing necessary for this study, commodity streaming multiprocessors were used; we demonstrate for structured meshes up to a 50 speed up in the computational time over traditional CPU implementations of the SIAC filter.
1 Introduction and Motivation
Smoothness-increasing accuracy-conserving (SIAC) filtering has demonstrated its
effectiveness in raising the convergence rate for discontinuous Galerkin solutions from order k + 21 to
order 2k + 1 for specific types of translation invariant meshes [79]. Additionally it improves
the weak continuity in the discontinuous Galerkin method to k 1 continuity. Typically this
improvement has a positive impact on the error quantity in the sense that it also reduces
the absolute errors in the solution. However, not enough emphasis has been placed on the
difference between superconvergent accuracy and improved errors. This distinction is
particularly important when it comes to interpreting the interplay introduced through meshing,
between geometry and filtering. The underlying mesh over which the DG solution is built is
important because the tool used in SIAC filteringconvolutionis scaled by the geometric
mesh size. This scaling heavily contributes to the effectiveness of the post-processor.
Although the choice of this scaling is straightforward when dealing with a uniform mesh, it is
not clear what the impact of either a global or local scaling will be on either the absolute
error or on the superconvergence properties of the post-processor. In this paper, we present
a study of the mesh scaling used in the SIAC filter and how it factors into the theoretical
errors. To accomplish the large volume of post-processing necessary for this study, commodity
streaming multiprocessors in the form of graphical processing units (GPUs) were used; we
demonstrate that when applied to structured meshes, up to a 50 speed up in the
computational time over traditional CPU implementations of the SIAC filter can be achieved. This
shows that it is feasible for SIAC filtering to be inserted into the post-processing pipeline as
a natural stage between simulation and further evaluation such as visualization.
The typical application of SIAC filters has been to discontinuous Galerkin solutions
on translation invariant meshes. The most typical means of generating translation
invariant meshes is by constructing a base tessellation of size H and repeatedly tiling in a
nonoverlapping fashion the base tessellation until the volume of interest is filled [1, 2]. The
effectiveness of such a translation invariant filter for discontinuous Galerkin solutions of
linear hyperbolic equations was initially demonstrated by Cockburn, Luskin, Shu and Sli [7].
A computational extension to smoothly-varying meshes as well as random meshes, where a
scaling equal to the largest element size was used, was given in [8]. For smoothly-varying
meshes, the improvement to order 2k + 1 was observed. For random meshes there was no
clear order improvement, which could be due to an incorrect kernel scaling. These results
were theoretically and numerically extended to translation invariant structured triangular
meshes in [9]. However, the outlook for triangular meshes is actually much better than those
presented in [9]. Indeed, the order improvement was not clear for filtering over a
UnionJack mesh when a filter scaling equal H2 was used (see Fig. 11). In this paper, we revisit
the Union-Jack mesh case, as well as a Chevron triangular mesh and demonstrate that it is
indeed (...truncated)