One-Sided Position-Dependent Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering Over Uniform and Non-uniform Meshes

In this paper, we introduce a new position-dependent smoothness-increasing accuracy-conserving (SIAC) filter that retains the benefits of position dependence as proposed in van Slingerland et al. (SIAM J Sci Comput 33:802–825, 2011) while ameliorating some of its shortcomings. As in the previous position-dependent filter, our new filter can be applied near domain boundaries, near a discontinuity in the solution, or at the interface of different mesh sizes; and as before, in general, it numerically enhances the accuracy and increases the smoothness of approximations obtained using the discontinuous Galerkin (dG) method. However, the previously proposed position-dependent one-sided filter had two significant disadvantages: (1) increased computational cost (in terms of function evaluations), brought about by the use of 4k + 1 central B-splines near a boundary (leading to increased kernel support) and (2) increased numerical conditioning issues that necessitated the use of quadruple precision for polynomial degrees of k ≥ 3 for the reported accuracy benefits to be realizable numerically. Our new filter addresses both of these issues—maintaining the same support size and with similar function evaluation characteristics as the symmetric filter in Jennifer K. Ryan, Xiaozhou Li: Supported by the Air Force Office of Scientific Research (AFOSR), Air Force Material Command, USAF, under Grant No. FA8655-13-1-3017. Robert M. Kirby: Supported by the Air Force Office of Scientific Research (AFOSR), Computational Mathematics Program (Program Manager: Dr. Fariba Fahroo), under Grant No. FA9550-08-1-0156. J. K. Ryan (B) School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK e-mail: [email protected] X. Li · K. Vuik Delft Institute of Applied Mathematics, Delft University of Technology, 2628 CD Delft, The Netherlands e-mail: [email protected] R. M. Kirby School of Computing, University of Utah, Salt Lake City, UT, USA e-mail: [email protected] K. Vuik e-mail: [email protected]