Abstract
The discontinuous Galerkin (DG) method has become popular in Computational Fluid Dynamics mainly due to its ability to achieve high-order solution accuracy on arbitrary grids, its high arithmetic intensity (measured as the ratio of the number of floating point operations to memory references), and the use of a local stencil that makes scalable parallel solutions possible. Despite its advantages, several difficulties hinder widespread use of the DG method, especially in industrial applications. One of the major challenges remaining is the capturing of discontinuities in a robust and accurate way. In our previous work, we have proposed a simple shock detector to identify discontinuities within a flow solution. The detector only utilizes local information to sense a shock/discontinuity ensuring that one of the key advantages of DG methods, their data locality, is not lost in transonic and supersonic flows. In this work, we reexamine the shock detector capabilities to distinguish between smooth and discontinuous solutions. Furthermore, we optimize the functional relationships between the shock detector and the filter strength, and present it in detail for others to use. By utilizing the shock detector and the corresponding filtering-strength relationships, one can robustly and accurately capture discontinuities ranging from very weak to strong shocks. Our method is demonstrated in a number of two-dimensional canonical examples.
Highlights
Ever since Computational Fluid Dynamics (CFD) began to play an important role in analyzing fluid motion and designing industrial products, engineers have sought techniques which can increase the accuracy of a flow simulation without increasing its associated computational cost
We proposed an approach for discontinuity detection for the discontinuous Galerkin (DG) method and demonstrated its shock-capturing capabilities [27]
To obtain the optimum accuracy from the DG discretization, the shock detector should not flag any element in the domain as a troubled element, as the solution to this problem is smooth
Summary
Ever since Computational Fluid Dynamics (CFD) began to play an important role in analyzing fluid motion and designing industrial products, engineers have sought techniques which can increase the accuracy of a flow simulation without increasing its associated computational cost. The DG method has not yet been fully adopted in industry mainly because of some remaining challenges for practical use One of these difficulties is the robust and accurate capturing of discontinuities within a flow solution, which is the main objective of the present work. It was not clear whether the preliminary relationships used provided excessive filtering which, despite its solution smoothness advantages, might lead to a severe loss in accuracy For this reason, in this paper the relationships between the shock detector and the filtering strengths are optimized to obtain maximum accuracy while maintaining robustness of the shock-capturing capability.
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