On a One-Sided Post-Processing Technique for the Discontinuous Galerkin Methods

Abstract

In this paper we study a class of one-sided post-processing techniques to enhance the accuracy of the discontinuous Galerkin methods. The applications considered in this paper are linear hyperbolic equations, however the technique can be used for the solution to a discontinuous Galerkin method solving other types of partial differential equations, or more general approximations, as long as there is a higher order negative norm error estimate for the numerical solution. The advantage of the one-sided post-processing is that it uses information only from one side, hence it can be applied up to domain boundaries, a discontinuity in the solution, or an interface of different mesh sizes. This technique allows us to obtain an improvement in the order of accuracy from k+1 of the discontinuous Galerkin method to 2k+1 of the post-processed solution, using piecewise polynomials of degree k, throughout the entire domain and not just away from the boundaries, discontinuities, or interfaces of different mesh sizes.

One-sided Post-processing for the Discontinuous Galerkin Method Using ENO Type Stencil Choosing and the Local Edge Detection Method

“…The applications of the post-processor were extended in [19] to include multi-domains with different mesh sizes as well as variable and discontinuous coefficient equations. The extension of this technique to one-sided post-processing appears in [20]. Detailed discussion of the post-processor, including implementation, can be found in [7,8,19,20].…”

“…The extension of this technique to one-sided post-processing appears in [20]. Detailed discussion of the post-processor, including implementation, can be found in [7,8,19,20].…”

“…The post-processing technique in [20] both reduces oscillations as well as improves accuracy in the L 2 norm. In fact it is possible to enhance the overall accuracy to be up to the order of the error estimate in the negative-order norm.…”

“…where ψ (k+1) (x) is obtained by convolving the characteristic function over the interval (− 1 2 , 1 2 ) with itself k times and c 2(k+1),k+1 γ ∈ R. The calculation of c 2(k+1),k+1 γ as well as the construction and implementation of (2.3) is described in [3,19,20,22,23].…”

“…In a previous paper by Ryan and Shu [20], a one-sided post-processing technique for the discontinuous Galerkin method was introduced for reconstructing solutions near computational boundaries and discontinuities in the solution, as well as for changes in mesh size. The technique requires a priori knowledge of shock locations, since no biasing should occur in smooth regions.…”