Jennifer K. Ryan

Extension of a Post Processing Technique for the Discontinuous Galerkin Method for Hyperbolic Equations with Application to an Aeroacoustic Problem

In this paper we further explore a local postprocessing technique, originally developed by Bramble and Schatz [Math. Comp., 31 (1977), pp. 94--111] using continuous finite element methods for elliptic problems and later by Cockburn et al. [Math. Comp., 72 (2003), pp. 577--606] using discontinuous Galerkin methods for hyperbolic equations. We investigate the technique in the context of superconvergence of the derivatives of the numerical solution, two space dimensions for both tensor product local basis and the usual kth degree polynomials basis, multidomain problems with different mesh sizes, variable coefficient linear problems including those with discontinuous coefficients, and linearized Euler equations applied to an aeroacoustic problem. We demonstrate through extensive numerical examples that the technique is very effective in all these situations in enhancing the accuracy of the discontinuous Galerkin solutions.

Investigation of Smoothness-Increasing Accuracy-Conserving Filters for Improving Streamline Integration through Discontinuous Fields

Streamline integration of fields produced by computational fluid mechanics simulations is a commonly used tool for the investigation and analysis of fluid flow phenomena. Integration is often accomplished through the application of ordinary differential equation (ODE) integrators-integrators whose error characteristics are predicated on the smoothness of the field through which the streamline is being integrated, which is not available at the interelement level of finite volume and finite element data. Adaptive error control techniques are often used to ameliorate the challenge posed by interelement discontinuities. As the root of the difficulties is the discontinuous nature of the data, we present a complementary approach of applying smoothness-increasing accuracy-conserving filters to the data prior to streamline integration. We investigate whether such an approach applied to uniform quadrilateral discontinuous Galerkin (high-order finite volume) data can be used to augment current adaptive error control approaches. We discuss and demonstrate through a numerical example the computational trade-offs exhibited when one applies such a strategy.

Postprocessing for the Discontinuous Galerkin Method over Nonuniform Meshes

A postprocessing technique based on negative order norm estimates for the discontinuous Galerkin methods was previously introduced by Cockburn, Luskin, Shu, and Suli [Proceedings of the International Symposium on Discontinuous Galerkin Methods, Springer, New York, pp. 291-300; Math. Comput., 72 (2003), pp. 577-606]. The postprocessor allows improvement in accuracy of the discontinuous Galerkin method for time-dependent linear hyperbolic equations from order

Smoothness-Increasing Accuracy-Conserving (SIAC) Postprocessing for Discontinuous Galerkin Solutions over Structured Triangular Meshes

k

Accuracy-enhancement of discontinuous Galerkin solutions for convection-diffusion equations in multiple-dimensions

+1 to order 2

Local derivative post-processing for the discontinuous Galerkin method

k

Multiwavelet troubled-cell indicator for discontinuity detection of discontinuous Galerkin schemes

+1 over a uniform mesh. Assumptions on the convolution kernel along with uniformity in mesh size give a local translation invariant postprocessor that allows for simple implementation using small matrix-vector multiplications. In this paper, we present two alternatives for extending this postprocessing technique to include smoothly varying meshes. The first method uses a simple local

Efficient Implementation of Smoothness-Increasing Accuracy-Conserving (SIAC) Filters for Discontinuous Galerkin Solutions

L^2

Position-Dependent Smoothness-Increasing Accuracy-Conserving (SIAC) Filtering for Improving Discontinuous Galerkin Solutions

-projection of the smoothly varying mesh to a locally uniform mesh and uses this projected solution to compute the postprocessed solution. By using this local

One-Sided Smoothness-Increasing Accuracy-Conserving Filtering for Enhanced Streamline Integration through Discontinuous Fields

L^2