Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Journal of Computational Physics
On one-sided filters for spectral Fourier approximations of discontinuous functions
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics - Orthogonal polynomials and numerical methods
Total variation diminishing Runge-Kutta schemes
Mathematics of Computation
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
Enhanced accuracy by post-processing for finite element methods for hyperbolic equations
Mathematics of Computation
SIAM Journal on Scientific Computing
IEEE Transactions on Visualization and Computer Graphics
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
Negative-Order Norm Estimates for Nonlinear Hyperbolic Conservation Laws
Journal of Scientific Computing
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Position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering is a promising technique not only in improving the order of the numerical solution obtained by a discontinuous Galerkin (DG) method but also in increasing the smoothness of the field and improving the magnitude of the errors. This was initially established as an accuracy enhancement technique by Cockburn et al. for linear hyperbolic equations to handle smooth solutions [Math. Comp., 72 (2003), pp. 577-606]. By implementing this technique, the quality of the solution can be improved from order $k+1$ to order $2k+1$ in the $L^2$-norm. Ryan and Shu used these ideas to extend this technique to be able to handle postprocessing near boundaries as well as discontinuities [Methods Appl. Anal., 10 (2003), pp. 295-307]. However, this presented difficulties as the resulting error had a stair-stepping effect and the errors themselves were not improved over those of the DG solution unless the mesh was suitably refined. In this paper, we discuss an improved filter for enhancing DG solutions that easily switches between one-sided postprocessing to handle boundaries or discontinuities and symmetric postprocessing for smooth regions. We numerically demonstrate that the magnitude of the errors using the modified postprocessor is roughly the same as that of the errors for the symmetric postprocessor itself, regardless of the boundary conditions.