A p-weighted limiter for the discontinuous Galerkin method on one-dimensional and two-dimensional triangular grids
Introduction
The discontinuous Galerkin (DG) method is widely used in compressible flow simulations for its attractive features such as arbitrary high-order accuracy, excellent parallel efficiency and easiness to accommodate arbitrary hp adaptivity [1]. Whereas a robust, accuracy-preserving and parameter-free shock capturing technique of the DG method still remains a challenge. There are successful shock capturing approaches for the finite difference (FD) and finite volume (FV) methods, while their extensions to the DG method are difficult. The difficulties are caused by some intrinsic properties of DG, such as the multiple degrees of freedoms (DOFs) inside a single element and the volume integral term after integration by parts. Over the last two decades, a great amount of methods has been developed to improve the efficiency and robustness of the shock capturing for the DG method.
There are two main kinds of shock capturing approaches for the DG method, i.e., solution limiting and artificial viscosity. Slope limiters, such as TVB limiter [2] and Barth's limiter [3], apply a nonlinear operator to the solution gradients to control spurious oscillations. The slope limiters are usually used together with a troubled-cell indicator to avoid significant accuracy degradation in smooth regions. Dumbser et al. [4] improved the accuracy and resolution of the DG method by applying a posterior troubled-cell indicator and updating sub-cell averages using the first/second-order TVD FV schemes. Ray and Hesthaven [5], [6] used an artificial neural network as the parameter-free troubled-cell indicator to improve the accuracy of the detection of discontinuities. WENO limiters achieved great success in the FD and FV frameworks [7], [8], [9], [10], [11], [12] and were generalized to the DG scheme. Zhu et al. [13] applied the FV type WENO limiter with a TVB detector on unstructured grids to the DG method, whereas the limiter required a large stencil and thus destroyed the compactness of the DG method. The Hermite WENO [14], [15] and secondary-reconstruction based WENO limiters [16], [17], [18] were developed to keep the compactness of the DG method. While for these WENO limiters, robust shock capturing, sub-cell resolution and extensions to high-order schemes () and high-order mesh still remain challenges.
Compared with solution limiting, the artificial viscosity approach has the advantages of higher sub-cell resolution, easier implementation on high-order mesh and better convergence rate. The challenge of the artificial viscosity method is the control of the amount of dissipation added to the governing equations. Persson and Peraire [19] determined the amount of viscosity required for stability by the resolution of the approximating space and achieved sub-cell shock capturing. Barter and Darmofal [20] proposed a smooth, PDE-based artificial viscosity model to avoid oscillations caused by element-to-element variations of the non-smooth artificial viscosity. Lv et al. [21] combined an entropy-residual shock detector with an artificial viscosity model for shock stabilization in the entropy-bounded discontinuous Galerkin framework.
The objective of this paper is to design a high-order limiter for the DG method fulfilling the following requirements: (1) accuracy preserving in smooth regions; (2) oscillation-free shock capturing in the sub-cells; (3) computationally efficient. Based on these motivations, a p-weighted limiter is proposed based on our previous work on the second-order limiter [22]. In the p-weighted limiter, the candidate polynomials are the p-hierarchical orthogonal polynomials of the current cell and the linear polynomials constructed using the projection method on the face-neighboring cells. Thus the candidate polynomials are of different orders, which is the essential difference between the p-weighted limiter and traditional WENO limiters. This difference provides the p-weighted limiter with a possibility that the solution can be degraded from a high-order polynomial in smooth regions to a linear one near shock waves. Therefore, the spurious numerical oscillations in the sub-cells near strong discontinuities could be well suppressed for the DG method.
The algorithm in computing the WENO weights is improved towards an accuracy-preserving p-weighted limiter. In the p-weighted procedure, the smoothness indicator needs to be of the same order of magnitude for all candidate polynomials in smooth regions to be accuracy-preserving. Whereas the traditional WENO smoothness indicator (SI) [7], [8] is designed for candidate polynomials of the same order. On coarse mesh, the value of the traditional WENO SI for low-order candidate polynomials is much smaller than that for high-order ones, which would result in accuracy degradation in smooth regions. A new smoothness indicator is proposed to predict close smoothness for the candidate polynomials of different orders. Furthermore, the new smoothness indicator is evaluated by a quadrature-free approach which takes advantage of the orthogonal property of the basis functions. In computing the WENO weights, the small positive number ϵ is set as a function of the smoothness indicator instead of a constant, to preserve accuracy in the vicinity of extremas.
The rest of this paper is organized as follows. Section 2 describes the framework of DG method for inviscid compressible flow simulation. Section 3 and 5 present the p-weighted limiter on 1D and 2D triangular grids. Numerical results are given in Section 4 and 6 to demonstrate the accuracy, resolution and robustness of the DG method using the p-weighted limiter.
Section snippets
The DG method for inviscid compressible flow simulation
This section presents the framework of the DG method for inviscid compressible flow simulation. The two-dimensional Euler equations in conservative form are where is the vector of conserved variables, with being the density, velocity, pressure and total energy, respectively. The inviscid flux F is well known and thus omitted here. The law for pressure of perfect gas is employed for closure of the Euler equations. Assuming
The p-weighted limiter on 1D grids
A high-resolution second-order WENO limiter was developed for the high-order DG spectral element method on mixed grids in [22]. A projection approach was proposed to construct the linear candidate polynomials. An accurate troubled-cell indicator is required to detect the very narrow regions near discontinuities. If the cells near extrema are marked as troubled cells in smooth regions, the second-order limiter will significantly degrade the order of accuracy of the original method.
The
Numerical results in 1D
Several one-dimensional benchmark test cases are solved to assess the accuracy-preserving and shock capturing capability of the p-weighted limiter. Uniform grids are used for all test cases. The CFL number is taken as 0.8 for all cases. Only for the blast wave problems in Section 4.4, the positivity-preserving strategy in [29] is adopted after limiting to avoid negative pressure and density. The limiter is applied on all cells to test the accuracy-preserving property. In all the plots, the
Extension to triangular grids
The extension of the p-weighted limiter from one-dimensional grids to two-dimensional triangular grids is not straightforward. First, the approximate solution in Eq. (3) uses the Lagrange interpolating bases, which should be converted to hierarchical orthogonal bases for easy construction of candidate polynomials. Second, for a triangular cell, there is no specific normal direction to compute the Jacobian matrix for the characteristic limiting. The characteristic limiting
Numerical results in 2D
In this section, several two-dimensional benchmark cases are used to test the accuracy-preserving and shock-capturing capabilities of the p-weighted limiter. The limiter is applied to all the cells to assess the accuracy of the limiter in smooth regions. To avoid negative density or pressure, the positivity-preserving strategy in [29] is applied after limiting. For the results of a p-th order DG scheme, each cell is divided into uniform sub-cells to generate contours in the sub-cell scale.
Conclusions
An accuracy-preserving p-weighted limiter is proposed for shock-capturing of the DG method on 1D and triangular grids. Unlike the traditional WENO limiters, the p-weighted limiter performs the weighted summation of the polynomials of different orders in the central cell and the linear ones constructed through the projection on the face-neighboring cells. The WENO smoothness indicator is modified to provide better smoothness assessment of the hierarchical candidate polynomials. A novel
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The work is financially supported by National Natural Science Foundation of China (Grant No. 11872383 and 11402313).
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