Positivity-preserving Discontinuous Galerkin Research Articles

A bound- and positivity-preserving discontinuous Galerkin method for solving the γ-based model

In this work, a bound- and positivity-preserving quasi-conservative discontinuous Galerkin (DG) method is proposed for the γ-based model of compressible two-medium flows. The contribution of this paper mainly includes three parts. On one hand, the DG method with the extended Harten-Lax-van Leer contact flux is proposed to solve the γ-based model, and satisfies the equilibrium-preserving property which preserves uniform velocity and pressure fields at an isolated material interface. On the other hand, an affine-invariant weighted essentially non-oscillatory (Ai-WENO) limiter is adopted to suppress oscillations near the discontinuities. The limiter with the Ai-WENO reconstruction method to the conservative variables not only is able to maintain the equilibrium property, but also generates sharper results around the locations of shock waves in contrast to that applying to the primitive variables. Last but not least, a flux-based bound- and positivity-preserving limiting strategy is introduced and analyzed, which preserves the physical bounds for auxiliary variables in the non-conservative governing equations, and the positivity for density and internal energy. Extensive numerical experiments in both one and two space dimensions show that the proposed method performs well in simulating compressible two-medium flows with high-order accuracy, equilibrium-preserving and bound-preserving properties.

Positivity-preserving discontinuous Galerkin scheme for linear hyperbolic equations with characteristics-informed augmentation

This paper presents a positivity-preserving discontinuous Galerkin (DG) scheme for the linear hyperbolic problem with variable coefficients on structured Cartesian domains. The standard DG spaces are augmented with either polynomial or non-polynomial basis functions. The primary purpose of these augmented basis functions is to ensure that the cell average from the unmodulated DG scheme remains positive. We explicitly obtain suitable basis functions by inspecting the method of characteristics on an auxiliary problem. A key result is proved which demonstrates that the unmodulated augmented DG scheme will retain a positive cell average, provided that the inflow, source term, and variable coefficients are positive. A simple scaling limiter can then be leveraged to produce a high-order conservative positivity-preserving DG scheme. Numerical experiments demonstrate the scheme is able to retain high-order accuracy as well as robustness for variable coefficients. To improve efficiency, an inexact augmented basis function can be obtained rather than a analytic non-polynomial solution to the auxiliary problem from the method of characteristics.

Positivity-Preserving Discontinuous Galerkin Methods on Triangular Meshes for Macroscopic Pedestrian Flow Models

The macroscopic models for solving the pedestrian flow problem can be generally classified into two categories as follows: first-order continuum models and high-order continuum models. In first-order continuum models, the density satisfies the mass conservation law, the speed is defined by a flow-density relationship, and the desired directional motion of pedestrians is determined by an Eikonal-type equation. In contrast, in high-order models, the velocity is governed by the momentum conservation law. In this study, we summarize existing first-order and high-order models and rewrite them in the form of unified scalar or system hyperbolic conservation laws. Next, we apply high-order discontinuous Galerkin methods with a positivity-preserving limiter on unstructured triangular meshes to solve the conservation law and a second-order fast-sweeping scheme to solve the Eikonal equations. Our method can efficiently model real-life complex computational regions and avoid nonphysical solutions and simulation blow-ups. Finally, numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed solution algorithm. The numerical results validate the reliability of the proposed numerical method and highlight the advantages of triangular meshes.

On the conservation property of positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations

Recently, there has been a series of works on the positivity-preserving discontinuous Galerkin methods for stationary hyperbolic equations, where the notion of mass conservation follows from a straightforward analogy of that of time-dependent problems, i.e. conserving the mass = preserving cell averages during limiting. Based on such a notion, the implementations and theoretical proofs of positivity-preserving limited methods for stationary equations are unnecessarily complicated and constrained. As will be shown in this paper, in some extreme cases, their convergence could even be problematic. In this work, we clarify a more appropriate definition of mass conservation for limiters applied to stationary hyperbolic equations and establish the genuinely conservative high-order positivity-preserving limited discontinuous Galerkin methods based on this definition. The new methods are able to preserve the positivity of solutions of scalar linear equations and scalar nonlinear equations with invariant wind direction, with much simpler implementations and easier proofs for accuracy and the Lax-Wendroff theorem, compared with the existing methods. Two types of positivity-preserving limiters preserving the local mass of stationary equations are developed to accommodate for the new definition of conservation and their accuracy are investigated. We would like to emphasize that a major advantage of the original DG scheme presented in [24] is a sweeping procedure, which allows for the computation of conservative steady-state solutions explicitly, cell by cell, without iterations, even for nonlinear equations as long as the wind direction is fixed. The main contribution of this paper is to introduce a limiting procedure to enforce positivity without changing the conservative property of this original DG scheme. The good performance of the algorithms for stationary hyperbolic equations and their applications in time-dependent problems are demonstrated by ample numerical tests.

A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

A High Order Positivity-Preserving Discontinuous Galerkin Remapping Method Based on a Moving Mesh Solver for ALE Simulation of the Compressible Fluid Flow

Numerical Modelling of a One-dimensional Dam Break Using a Slope-Limiting Positivity-Preserving Discontinuous Galerkin Method

The shallow water equations have been widely used to simulate dam-break problem. Due to the nonlinear and discontinuous nature of the event, application of numerical methods is often plagued with problem such as oscillation and unphysical solutions. The aim of this paper is to simulate a one-dimensional dam break using an improved Discontinuous Galerkin Method. The governing equation is discretized using linear shape function and explicit third-order Runge-Kutta. Problem with negative water depth due to wetting and drying is treated using a combination of thin water layer and min-mod slope limiting techniques. The use of these techniques prevents solution blow-out and oscillation especially at the wet-dry interface. Comparison with available analytical solution shows excellent agreement without under and over-shooting of the solution. Convergence study demonstrate that the overall accuracy of the method is consistent with the shape function order. The method is promising as it can easily be extended to higher order and can be used as an alternative to typical method such as Finite Volume.

High order conservative positivity-preserving discontinuous Galerkin method for stationary hyperbolic equations

This is a follow-up work of Yuan et al. (2016) [19] and Ling et al. (2018) [13] that further investigates the positivity-preserving discontinuous Galerkin (DG) methods for stationary hyperbolic equations. In 2016, Yuan et al. proposed a high order positivity-preserving DG method for stationary hyperbolic equations with constant coefficients, but the scheme has to be used in combination with a non-conservative rotational limiter introduced in case of negative cell averages. Ling et al. (2018) improved the results in one dimensional space by rigorously proving the positivity of cell averages of the unmodulated DG scheme, which allows the conservative scaling limiter in Zhang and Shu (2010) [22] to be used to maintain positivity without affecting accuracy, but extension to two space dimensions requires an augmentation of the DG space and works only in the second order case. Considering that the aforementioned works only address stationary hyperbolic equations with constant coefficients and higher than second order conservative methods are still unavailable in two and three space dimensions, we propose high order conservative positivity-preserving DG methods for variable coefficient and nonlinear stationary hyperbolic equations in one dimension and constant coefficient stationary hyperbolic equations in two and three dimensions, via a suitable quadrature in the DG framework. We show the good performance of the algorithms by ample numerical experiments, including their applications in time-dependent problems.

High order entropy stable and positivity-preserving discontinuous Galerkin method for the nonlocal electron heat transport model

The nonlocal electron heat transport model in laser heated plasmas plays a crucial role in inertial confinement fusion (ICF), and it is important to solve it numerically in an accurate and robust way. In this paper, we first develop an one-dimensional high-order entropy stable discontinuous Galerkin method for the nonlocal electron heat transport model. We further design our DG scheme to have the positivity-preserving property by using the scaling positivity-preserving limiter, which is shown, by a computer-aided proof, to have no extra time step constraint than that required by L2 stability. Next, we extend our one-dimensional scheme to two-dimensions on rectangular meshes and tensor product polynomial spaces. Numerical examples are given to verify the high-order accuracy and positivity-preserving properties of our scheme. By comparing the local and nonlocal electron heat transport models, we also observe more physical phenomena such as the flux reduction and the preheat effect from the nonlocal model.

Positivity-preserving third order DG schemes for Poisson–Nernst–Planck equations

In this paper, we design and analyze third order positivity-preserving discontinuous Galerkin (DG) schemes for solving the time-dependent system of Poisson–Nernst–Planck (PNP) equations, which have found much use in diverse applications. Our DG method with Euler forward time discretization is shown to preserve the positivity of cell averages at all time steps. The positivity of numerical solutions is then restored by a scaling limiter in reference to positive weighted cell averages. The method is also shown to preserve steady states. Numerical examples are presented to demonstrate the third order accuracy and illustrate the positivity-preserving property in both one and two dimensions.

Entropy stabilization and property-preserving limiters for ℙ1 discontinuous Galerkin discretizations of scalar hyperbolic problems

Abstract The methodology proposed in this paper bridges the gap between entropy stable and positivity-preserving discontinuous Galerkin (DG) methods for nonlinear hyperbolic problems. The entropy stability property and, optionally, preservation of local bounds for cell averages are enforced using flux limiters based on entropy conditions and discrete maximum principles, respectively. Entropy production by the (limited) gradients of the piecewise-linear DG approximation is constrained using Rusanov-type entropy viscosity. The Taylor basis representation of the entropy stabilization term reveals that it penalizes the solution gradients in a manner similar to slope limiting and requires implicit treatment to avoid severe time step restrictions. The optional application of a vertex-based slope limiter constrains the DG solution to be bounded by local maxima and minima of the cell averages. Numerical studies are performed for two scalar two-dimensional test problems with nonlinear and nonconvex flux functions.

Numerical study of non-uniqueness for 2D compressible isentropic Euler equations

In this paper, we numerically study a class of solutions with spiraling singularities in vorticity for two-dimensional, inviscid, compressible Euler systems, where the initial data have an algebraic singularity in vorticity at the origin. These are different from the multi-dimensional Riemann problems widely studied in the literature. Our computations provide numerical evidence of the existence of initial value problems with multiple solutions, thus revealing a fundamental obstruction toward the well-posedness of the governing equations. The compressible Euler equations are solved using the positivity-preserving discontinuous Galerkin method.

Entropy Stable and Well-Balanced Discontinuous Galerkin Methods for the Nonlinear Shallow Water Equations

The nonlinear shallow water equations (SWEs) are widely used to model the unsteady water flows in rivers and coastal areas, with extensive applications in ocean and hydraulic engineering. In this work, we propose entropy stable, well-balanced and positivity-preserving discontinuous Galerkin (DG) methods, under arbitrary choices of quadrature rules, for the SWEs with a non-flat bottom topography. In Chan (J Comput Phys 362:346–374, 2018), a SBP-like differentiation operator was introduced to construct the discretely entropy conservative DG methods. We extend this idea to the SWEs and establish an entropy stable scheme by adding additional dissipative terms. Careful approximation of the source term is included to ensure the well-balanced property of the resulting method. A simple positivity-preserving limiter, compatible with the entropy stable property, is included to guarantee the non-negative water heights during the computation. One- and two-dimensional numerical experiments are presented to demonstrate the performance of the proposed methods.

High order positivity-preserving discontinuous Galerkin schemes for radiative transfer equations on triangular meshes

It is an important and challenging issue for the numerical solution of radiative transfer equations to maintain both high order accuracy and positivity. For the two-dimensional radiative transfer equations, Ling et al. give a counterexample (Ling et al. (2018) [13]) showing that unmodulated discontinuous Galerkin (DG) solver based either on the Pk or Qk polynomial spaces could generate negative cell averages even if the inflow boundary value and the source term are both positive (and, for time dependent problems, also a nonnegative initial condition). Therefore the positivity-preserving frameworks in Zhang and Shu (2010) [28] and Zhang et al. (2012) [29] which are based on the value of cell averages being positive cannot be directly used to obtain a high order conservative positivity-preserving DG scheme for the radiative transfer equations neither on rectangular meshes nor on triangular meshes. In Yuan et al. (2016) [26], when the cell average of DG schemes is negative, a rotational positivity-preserving limiter is constructed which could keep high order accuracy and positivity in the one-dimensional radiative transfer equations with Pk polynomials and could be straightforwardly extended to two-dimensional radiative transfer equations on rectangular meshes with Qk polynomials (tensor product polynomials). This paper presents an extension of the idea of the above mentioned one-dimensional rotational positivity-preserving limiter algorithm to two-dimensional high order positivity-preserving DG schemes for solving steady and unsteady radiative transfer equations on triangular meshes with Pk polynomials. The extension of this method is conceptually plausible but highly nontrivial. We first focus on finding a special quadrature rule on a triangle which should satisfy some conditions. The most important one is that the quadrature points can be arranged on several line segments, on which we can use the one-dimensional rotational positivity-preserving limiter. Since the number of the quadrature points is larger than the number of basis functions of Pk polynomial space, we determine a k-th polynomial by a L2-norm Least Square subject to its cell average being equal to the weighted average of the values on the quadrature points after using the rotational positivity-preserving limiter. Since the weights used here are the quadrature weights which are positive, then the cell average of the modified polynomial is nonnegative. And the final modified polynomial can be obtained by using the two-dimensional scaling positivity-preserving limiter on the triangular element. We theoretically prove that our rotational positivity-preserving limiter on triangular meshes could keep both high order accuracy and positivity. It is relatively simple to implement, and also does not affect convergence to weak solutions. The numerical results validate the high order accuracy and the positivity-preserving properties of our schemes. The advantage of the triangular meshes on handling complex domain is also presented in our numerical examples.

Conservative High Order Positivity-Preserving Discontinuous Galerkin Methods for Linear Hyperbolic and Radiative Transfer Equations

We further investigate the high order positivity-preserving discontinuous Galerkin (DG) methods for linear hyperbolic and radiative transfer equations developed in Yuan et al. (SIAM J Sci Comput 38:A2987–A3019, 2016). The DG methods in Yuan et al. (2016) can maintain positivity and high order accuracy, but they rely both on the scaling limiter in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and a rotational limiter, the latter may alter cell averages of the unmodulated DG scheme, thereby affecting conservation. Even though a Lax–Wendroff type theorem is proved in Yuan et al. (2016), guaranteeing convergence to weak solutions with correct shock speed when such rotational limiter is applied, it would still be desirable if a conservative DG method without changing the cell averages can be obtained which has both high order accuracy and positivity-preserving capability. In this paper, we develop and analyze such a DG method for both linear hyperbolic equations and radiative transfer equations. In the one-dimensional case, the method uses traditional DG space $$P^k$$ of piecewise polynomials of degree at most k. A key result is proved that the unmodulated DG solver in this case can maintain positivity of the cell average if the inflow boundary value and the source term are both positive, therefore the positivity-preserving framework in Zhang and Shu (2010) can be used to obtain a high order conservative positivity-preserving DG scheme. Unfortunately, in two-dimensions this is no longer the case. We show that the unmodulated DG solver based either on $$P^k$$ or $$Q^k$$ spaces (piecewise kth degree polynomials or piecewise tensor-product kth degree polynomials) could generate negative cell averages. We augment the DG space with additional functions so that the positivity of cell averages from the unmodulated DG solver can be restored, thereby leading to high order conservative positivity-preserving DG scheme based on these augmented DG spaces following the framework in Zhang and Shu (2010). Computational results are provided to demonstrate the good performance of our DG schemes.

A positivity-preserving high order discontinuous Galerkin scheme for convection–diffusion equations

For constructing high order accurate positivity-preserving schemes for convection–diffusion equations, we construct a simple positivity-preserving diffusion flux. Discontinuous Galerkin (DG) schemes with such a positivity-preserving diffusion flux are nonlinear schemes, which can be regarded as a reduction of the high order positivity-preserving DG schemes for compressible Navier–Stokes equations in [1] to scalar diffusion operators. In this paper we focus on the local DG method to discuss how to apply such a flux. A limiter on the auxiliary variable for approximating the gradient of the solution must be used so that the diffusion flux is positivity-preserving in the sense that DG schemes with this flux satisfies a weak positivity property. Together with a positivity-preserving limiter, high order DG schemes with strong stability preserving time discretizations can be rendered positivity-preserving without losing conservation or high order accuracy for convection–diffusion problems with periodic boundary conditions or a special class of Dirichlet or Neumann boundary conditions. Numerical tests on a few parabolic equations and an application to modeling electrical discharges are shown to demonstrate the performance of this scheme.

Implicit Positivity-Preserving High-Order Discontinuous Galerkin Methods for Conservation Laws

Positivity-preserving discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws have been extensively studied in the last several years, but nearly all the developed schemes are coupled with explicit time discretizations. Explicit discretizations suffer from the constraint for the Courant--Friedrichs--Lewy (CFL) number. This makes explicit methods impractical for problems involving unstructured and extremely varying meshes or long-time simulations. Instead, implicit DG schemes are often popular in practice, especially in the computational fluid dynamics (CFD) community. In this paper we develop a high-order positivity-preserving DG method with the backward Euler time discretization for conservation laws. We focus on one spatial dimension. However, the result easily generalizes to multidimensional tensor product meshes and polynomial spaces. This work is based on a generalization of the positivity-preserving limiters in [X. Zhang and C.-W. Shu, J. Comput. Phys., 229 (2010), pp. 3091--312...

A second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin scheme for the Kerr–Debye model

In this paper, we develop a second-order asymptotic-preserving and positivity-preserving discontinuous Galerkin (DG) scheme for the Kerr–Debye model. By using the approach first introduced by Zhang and Shu in [Q. Zhang and C.-W. Shu, Error estimates to smooth solutions of Runge–Kutta discontinuous Galerkin methods for scalar conservation laws, SIAM J. Numer. Anal. 42 (2004) 641–666.] with an energy estimate and Taylor expansion, the asymptotic-preserving property of the semi-discrete DG methods is proved rigorously. In addition, we propose a class of unconditional positivity-preserving implicit–explicit (IMEX) Runge–Kutta methods for the system of ordinary differential equations arising from the semi-discretization of the Kerr–Debye model. The new IMEX Runge–Kutta methods are based on the modification of the strong-stability-preserving (SSP) implicit Runge–Kutta method and have second-order accuracy. The numerical results validate our analysis.

Application of Positivity-Preserving Discontinuous Galerkin Scheme in Gaseous Detonations

In numerical simulation of gaseous detonation, due to the complexity of the computational domain, negative density and pressure often emerge in high resolution numerical computing, which leads to blow-ups. The paper provides high order discontinuous Galerkin (DG) positivity-preserving scheme for two-dimensional (2D) Euler equations with two-step chemical reaction which preserve positivity of density, pressure and chemical reaction process. A positivity-preserving limiter is added in high order DG scheme without influencing conservation, accuracy and stability. The method is verified by parallel numerical simulations and is approved to be well applied to numerical simulation of gaseous detonation propagation with complicated geometry boundary.

On positivity-preserving high order discontinuous Galerkin schemes for compressible Navier–Stokes equations

We construct a local Lax–Friedrichs type positivity-preserving flux for compressible Navier–Stokes equations, which can be easily extended to multiple dimensions for generic forms of equations of state, shear stress tensor and heat flux. With this positivity-preserving flux, any finite volume type schemes including discontinuous Galerkin (DG) schemes with strong stability preserving Runge–Kutta time discretizations satisfy a weak positivity property. With a simple and efficient positivity-preserving limiter, high order explicit Runge–Kutta DG schemes are rendered preserving the positivity of density and internal energy without losing local conservation or high order accuracy. Numerical tests suggest that the positivity-preserving flux and the positivity-preserving limiter do not induce excessive artificial viscosity, and the high order positivity-preserving DG schemes without other limiters can produce satisfying non-oscillatory solutions when the nonlinear diffusion in compressible Navier–Stokes equations is accurately resolved.

Positivity-Preserving Discontinuous Galerkin Methods with Lax–Wendroff Time Discretizations

This work introduces a single-stage, single-step method for the compressible Euler equations that is provably positivity-preserving and can be applied on both Cartesian and unstructured meshes. This method is the first case of a single-stage, single-step method that is simultaneously high-order, positivity-preserving, and operates on unstructured meshes. Time-stepping is accomplished via the Lax---Wendroff approach, which is also sometimes called the Cauchy---Kovalevskaya procedure, where temporal derivatives in a Taylor series in time are exchanged for spatial derivatives. The Lax---Wendroff discontinuous Galerkin (LxW-DG) method developed in this work is formulated so that it looks like a forward Euler update but with a high-order time-extrapolated flux. In particular, the numerical flux used in this work is a convex combination of a low-order positivity-preserving contribution and a high-order component that can be damped to enforce positivity of the cell averages for the density and pressure for each time step. In addition to this flux limiter, a moment limiter is applied that forces positivity of the solution at finitely many quadrature points within each cell. The combination of the flux limiter and the moment limiter guarantees positivity of the cell averages from one time-step to the next. Finally, a simple shock capturing limiter that uses the same basic technology as the moment limiter is introduced in order to obtain non-oscillatory results. The resulting scheme can be extended to arbitrary order without increasing the size of the effective stencil. We present numerical results in one and two space dimensions that demonstrate the robustness of the proposed scheme.