A p-weighted limiter for the discontinuous Galerkin method on one-dimensional and two-dimensional triangular grids

Abstract

This paper presents an accuracy-preserving p-weighted limiter for discontinuous Galerkin methods on one-dimensional and two-dimensional triangular grids. The p-weighted limiter is the extension of the second-order WENO limiter by Li et al. (2018) [22] to high-order accuracy, with the following important improvements of the limiting procedure. First, the candidate polynomials of the p-weighted limiter are the p-hierarchical orthogonal polynomials of the current cell, and the linear polynomials constructed by minimizing the projection error on the face-neighboring cells. Second, the p-weighted procedure introduces a new smoothness indicator which has less numerical dissipation comparing with the classical WENO one. The smoothness indicator is efficiently computed through a quadrature-free approach that takes advantage of the orthogonal property of the basis functions. Third, the small positive number ϵ, which is introduced in the weights to avoid dividing by zero, is set as a function of the smoothness indicator to preserve accuracy near smooth extremas. Numerous benchmark problems are solved to test the p1, p3 and p5 discontinuous Galerkin schemes using the p-weighted limiter. Numerical results demonstrate that the p-weighted limiter is capable of capturing strong shocks while preserving accuracy in smooth regions.

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 ($p>2$) 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 $L_2$ 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.