Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

doi:10.1016/j.cma.2021.113876

Computer Methods in Applied Mechanics and Engineering

Volume 382, 15 August 2021, 113876

https://doi.org/10.1016/j.cma.2021.113876 Get rights and content

Highlights

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High-order, robust, invariant domain preserving discontinuous Galerkin methods.
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New, efficient dimension-by-dimension subcell limiting procedure.
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Sparsified low-order discretization with compact stencils using graph viscosity.
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High-order convergence using subcell smoothness indicator.

Abstract

In this paper, we develop high-order nodal discontinuous Galerkin (DG) methods for hyperbolic conservation laws that satisfy invariant domain preserving properties using subcell flux corrections and convex limiting. These methods are based on a subcell flux corrected transport (FCT) methodology that involves blending a high-order target scheme with a robust, low-order invariant domain preserving method that is obtained using a graph viscosity technique. The new low-order discretizations are based on sparse stencils which do not increase with the polynomial degree of the high-order DG method. As a result, the accuracy of the low-order method does not degrade when used with high-order target methods. The method is applied to both scalar conservation laws, for which the discrete maximum principle is naturally enforced, and to systems of conservation laws such as the Euler equations, for which positivity of density and a minimum principle for specific entropy are enforced. Numerical results are presented on a number of benchmark test cases.

Introduction

High-order numerical methods have been successfully applied to a wide range of applications [1], [2], [3]. These methods promise higher accuracy per degree of freedom when compared with traditional low-order methods, and have the potential to achieve high efficiency on modern computing architectures [4], [5], [6]. For example, in the field of computational fluid dynamics, such methods have seen particular success when applied to under-resolved turbulent flows, such as in the context of implicit large eddy simulation (ILES) [7], [8], [9]. However, a critical issue that must be addressed when applying high-order methods to convection-dominated problems is their robustness, especially in the context of nonlinear problems with discontinuous features such as shock waves [10], [11], [12].

In particular, discontinuous Galerkin (DG) methods have seen considerable success when applied to convection dominated problems [13]. These methods possess many desirable properties, such as arbitrary formal order of accuracy, and suitability for use with unstructured meshes. The robustness of DG methods is the subject of a large body of research [7], [14]. For scalar conservation laws and symmetric systems, the DG method satisfies a cell entropy inequality [15]. However, for general hyperbolic systems, such as the Euler equations, techniques such as flux differencing are required to ensure entropy stability [16], [17], [18], [19]. Furthermore, the use of high degree polynomials can introduce oscillations, and therefore limiters or artificial viscosity techniques are often used for bounds preservation, monotonicity, and shock capturing [20], [21], [22], [23].

An alternative to the above stabilization and limiting strategies is an approach developed by Guermond, Popov, and colleagues, based on invariant domain preserving (IDP) discretizations and convex limiting [24], [25], [26], [27]. A desirable property for numerical discretizations of hyperbolic conservation laws is invariant domain preservation: if the exact solution to the conservation law lies in a convex invariant set, then the numerical solution should as well [24]. This is a generalization of the concept of a discrete maximum principle, and will ensure that the discretization is bounds preserving, positivity preserving, and non-oscillatory. Suitable low-order invariant domain preserving (IDP) discretizations have been paired with high-order discretizations using convex limiting or algebraic flux correction strategies to obtain second-order accurate methods that preserve specific invariant domains [25], [26], [27].

In this work, we develop high-order discontinuous Galerkin methods that satisfy invariant domain preserving properties using a convex limiting strategy. The limiting strategy makes use of a novel sparse low-order IDP method whose stencil does not grow with the polynomial degree of the corresponding high-order method. Crucially, the accuracy of the low-order method does not degrade as the polynomial degree of the high-order method is increased, as is observed to occur with more naive graph viscosity approaches. Related strategies for sparsifying the convective operator for Bernstein basis finite element methods were previously developed by Kuzmin and colleagues [28]. The flux-corrected method is obtained by performing an efficient, dimension-by-dimension subcell flux correction procedure, blending the low-order IDP method with the high-order target DG method. The resulting method is conservative, and can satisfy any number of constraints on quasiconcave functionals specified by the user (cf. [25]). Since the accuracy of the low-order method does not decrease with the polynomial degree of the target method, we observe more accurate results using higher-order methods with a fixed number of degrees of freedom, even on problems with discontinuous solutions. This method can also be combined with a subcell resolution smoothness indicator to alleviate peak clipping effects near smooth extrema.

The structure of this paper is as follows. In Section 2, we formulate the high-order DG discretization, and state some key properties. In Section 3, we introduce a new low-order sparsified discretization that can be rendered invariant domain preserving using a graph viscosity approach. We develop a subcell flux correction strategy for blending the high-order (target) method and the low-order IDP method in Section 4. As specific examples, applications to the linear advection equation with variable velocity field and the Euler equations of gas dynamics are discussed. A number of numerical test cases demonstrating the effectiveness of the method on both scalar equations and systems of hyperbolic conservation laws are presented in Section 5. Finally, we end with some concluding remarks in Section 6.

Section snippets

Governing equations and discretization

Consider a system of hyperbolic conservation laws, $\frac{\partial u}{\partial t} + \nabla \cdot F (u) = 0,$ with solution $u (x, t) \in R^{n_{c}}$ , $x \in Ω \subseteq R^{d}$ . The flux function is given by $F (u (x)) \in R^{d \times n_{c}}$ . The spatial dimension is denoted $d$ , and the number of solution components is $n_{c}$ . The initial conditions are given by $u (x, 0) = u_{0} (x)$ . Closely associated with the problem (1) is the following one dimensional Riemann problem, which will be important both for the definition of invariant domain preservation, and for the formulation of the discontinuous

Construction of the IDP low-order method

We now modify the discretization (13) with the goal of obtaining a method which is invariant domain preserving (IDP). In Guermond and Popov [24] this is done by adding a graph viscosity term that is based on the guaranteed maximum speed (GMS) of the hyperbolic system. One potential issue with this approach is that the amount of graph viscosity added to the discretization increases as the size of the discrete stencil increases. As a result, applying this technique to high-order methods with

Flux correction and limiting strategies

We now combine the low-order IDP discretization (33) with the high-order discretization (13), to obtain a bound-preserving scheme. To this end, we consider a forward Euler time discretization of both the low-order and high-order approximations (as mentioned above, the extension to high-order time integration is straightforward using SSP methods). The provisional high-order update for element $K$ is given by $\frac{m_{i}}{Δ t} u_{i}^{H, n + 1} = \frac{m_{i}}{Δ t} u_{i}^{n} - (\sum_{i = 1}^{d} D_{i, R} \sum_{j = 1}^{d} {G J}_{i j}^{- 1} F_{j}^{n} + \sum_{e \in \partial K} B_{e, R} ({\hat{F}}_{e}^{n} - F_{e}^{n} \cdot n)),$ and the low-order IDP

Numerical examples

The method was implemented in the MFEM finite element framework [46], and is tested on a variety of benchmark problems, including scalar problems and hyperbolic systems, in 1D and 2D. For these test cases, unless stated otherwise, we integrate in time using the third-order SSP Runge–Kutta method [47]. The time step is chosen according to (38), with a CFL constant of $1 ∕ 2$ , i.e. $Δ t = \frac{1}{2} {min}_{i} \frac{m_{i}}{2 {\hat{d}}_{i i}}$ .

Conclusions

In this work, we have presented a discontinuous Galerkin spectral element method with convex limiting for hyperbolic conservation laws. This method preserves any specified set of invariant domain properties (e.g. local maximum principles, positivity of pressure and density, minimum principle for specific entropy, etc.). The method is based on an efficient dimension-by-dimension subcell blending of the target high-order (unlimited) DG-SEM method, and a low-order, invariant domain preserving

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.

Acknowledgments

The author acknowledges H. Hajduk and D. Kuzmin for insightful conversations and comments on this work.

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory, USA under Contract DE-AC52-07NA27344. LLNL-JRNL-808645. This document was prepared as an account of work sponsored by an agency of the United States government . Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees

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Sparse invariant domain preserving discontinuous Galerkin methods with subcell convex limiting

Highlights

Abstract

Introduction

Section snippets

Governing equations and discretization

Construction of the IDP low-order method

Flux correction and limiting strategies

Numerical examples

Conclusions

Declaration of Competing Interest

Acknowledgments

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Appl. Numer. Math.

Comput. Methods Appl. Mech. Engrg.

Comput. Methods Appl. Mech. Engrg.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Comput. & Fluids

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

CFD Vision 2030 Study: a Path to Revolutionary Computational AerosciencesTechnical Report NASA/CR-2014-218178

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Internat. J. Numer. Methods Fluids

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