Discontinuous Galerkin Scheme Research Articles

An Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Scheme for Compressible Multi-Material Flows on Adaptive Quadrilateral Meshes

An Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Scheme for Compressible Multi-Material Flows on Adaptive Quadrilateral Meshes

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.

Approximately well-balanced Discontinuous Galerkin methods using bases enriched with Physics-Informed Neural Networks

This work concerns the enrichment of Discontinuous Galerkin (DG) bases, so that the resulting scheme provides a much better approximation of steady solutions to hyperbolic systems of balance laws. The basis enrichment leverages a prior – an approximation of the steady solution – which we propose to compute using a Physics-Informed Neural Network (PINN). To that end, after presenting the classical DG scheme, we show how to enrich its basis with a prior. Convergence results and error estimates follow, in which we prove that the basis with prior does not change the order of convergence, and that the error constant is improved. To construct the prior, we elect to use parametric PINNs, which we introduce, as well as the algorithms to construct a prior from PINNs. We finally perform several validation experiments on four different hyperbolic balance laws to highlight the properties of the scheme. Namely, we show that the DG scheme with prior is much more accurate on steady solutions than the DG scheme without prior, while retaining the same approximation quality on unsteady solutions.

Comparison of high-order numerical methodologies for the simulation of the supersonic Taylor–Green vortex flow

This work presents a comparison of several high-order numerical methodologies for simulating shock/turbulence interactions based on the supersonic Taylor–Green vortex flow, considering a Reynolds number of 1600 and a Mach number of 1.25. The numerical schemes considered include high-order Finite Difference, Targeted Essentially Non-Oscillatory, Discontinuous Galerkin, and Spectral Difference schemes. The shock capturing methods include high-order filtering, localized artificial diffusivity, non-oscillatory numerical fluxes, and local low-order switching. The ability of the various high-order numerical methodologies to both capture shocks and represent accurately the development of turbulent vortices is assessed.

High order well-balanced asymptotic preserving IMEX RKDG schemes for the two-dimensional nonlinear shallow water equations

In this paper, a high order well-balanced asymptotic preserving scheme is presented for the two-dimensional nonlinear shallow water equations over variable bottom topography in all Froude number regimes. To obtain the well-balanced property, the system is first reformulated as a new form by introducing an auxiliary parameter. The flux is then split into a linear stiff part to be treated implicitly and a nonlinear non-stiff part to be treated explicitly, and the source term is treated explicitly. An implicit-explicit Runge-Kutta discontinuous Galerkin scheme is designed for solving the equations. The proposed scheme can be proved to be well-balanced, asymptotic preserving and asymptotically accurate. Finally, several numerical tests are carried out to validate the performance of our proposed scheme.

Numerical Studies on the Classical Long Wave System Describing Shallow Water Waves With Dispersion

In this study, a reliable and efficient local discontinuous Galerkin scheme for numerically solving the classical long wave system is proposed. This scheme approximates the temporal derivatives using the explicit strong-stability-preserving higher-order Runge-Kutta method, while the space derivatives are approximated using the local discontinuous Galerkin method, yielding an ordinary differential equation system. Numerical examples for various test problems are presented to offer a comprehensive understanding of the accuracy and reliability of the proposed method. The results obtained, which confirm the sub-optimal order of accuracy, are displayed in multiple tables. Additionally, several two-dimensional and three-dimensional graphical depictions of the problem have been provided to illustrate the behavior of the solution.

On the stability and convergence of discontinuous Galerkin schemes for incompressible flows

Abstract The property that the velocity $\textbf{u}$ belongs to $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$ is an essential requirement in the definition of energy solutions of models for incompressible fluids. It is, therefore, highly desirable that the solutions produced by discretization methods are uniformly stable in the $L^{\infty }(0,T;L^{2}(\varOmega )^{d})$-norm. In this work, we establish that this is indeed the case for discontinuous Galerkin (DG) discretizations (in time and space) of non-Newtonian models with $p$-structure, assuming that $p\geq \frac{3d+2}{d+2}$; the time discretization is equivalent to the RadauIIA Implicit Runge–Kutta method. We also prove (weak) convergence of the numerical scheme to the weak solution of the system; this type of convergence result for schemes based on quadrature seems to be new. As an auxiliary result, we also derive Gagliardo–Nirenberg-type inequalities on DG spaces, which might be of independent interest.

A high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for compressible multi-material flows

In this paper, a high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) scheme is proposed for simulating one-dimensional compressible multi-material flows. The system is discretized by the DG method, and a kind of Taylor’s expansion basis functions in the general cell is used to construct the interpolation polynomials of the variables. The mesh velocity at the node recognized as the material interface is set to approximate fluid velocity, which makes our scheme can capture the material interface accurately and clearly. For controlling the oscillations and reducing the numerical dissipation, a reconstruction based on the boundary variation diminishing (BVD) algorithm and Tangent of Hyperbola for INterface Capturing (THINC) function is employed to minimize the jump between the values at the left and right sides of cell boundaries. Due to the essentially monotone and bounded properties of THINC function, the difficulties caused by solving discontinuous solutions and complexities of designing limiters can be avoided. Finally, several examples are presented to demonstrate the accuracy and good performance of our scheme such as the essentially non-oscillatory property and so on for simulating the compressible multi-material flows.

Comparison of different discontinuous Galerkin methods based on various reformulations for gKdV equation: Soliton dynamics and blowup

In this work, we explore the use of several discontinuous Galerkin (DG) methods for simulating soliton dynamics in the generalized Korteweg-de Vries (gKdV) equation. The presence of high-order nonlinearity in this model can lead to the finite-time blowup phenomenon, which challenges every aspect of a numerical scheme. The considered DG schemes encompass popular methods derived from the energy and/or Hamiltonian conservations of the gKdV equation. To evaluate the performance of these DG schemes from various angles, we present a series of numerical experiments. Through comprehensive comparisons, we aim to identify the scheme that exhibits the best performance. To further enhance the accuracy and efficiency of DG methods, particularly in the context of blowup simulations, we incorporate the arbitrary Lagrangian-Eulerian method for adaptive mesh movement.

A leap-frog nodal discontinuous Galerkin method for Maxwell polynomial chaos Debye model

The Maxwell polynomial chaos Debye model uses the parameter distribution in the Debye model to represent the Cole-Cole dispersive mechanism. In this paper, a leap-frog nodal discontinuous Galerkin (DG) method is studied for solving this model. Under the Courant-Friedrichs-Lewy (CFL) condition, the stability of the fully discrete DG scheme is demonstrated and its error estimate is derived. Numerical examples are provided for supporting the correctness of the theoretical analysis and the effectiveness of our proposed method.

A Posteriori Local Subcell Correction of High-Order Discontinuous Galerkin Scheme for Conservation Laws on Two-Dimensional Unstructured Grids

A Posteriori Local Subcell Correction of High-Order Discontinuous Galerkin Scheme for Conservation Laws on Two-Dimensional Unstructured Grids

A positivity preserving and oscillation-free entropy stable discontinuous Galerkin scheme for the reactive Euler equations

The reactive Euler equations are a basic model for fluid flows with chemical reactions. In this work, we construct a high order positivity preserving and oscillation-free entropy stable discontinuous Galerkin (DG) scheme for the reactive Euler equations. The main ingredients of the scheme include (i) entropy preserving and entropy stable fluxes to achieve entropy stability, (ii) artificial damping terms to restrain spurious oscillations near the shocks, and (iii) positivity preserving limiters to guarantee the positivity of solutions. These ingredients are compatible with each other so that our scheme simultaneously enjoys the properties of entropy stable, oscillation-free and positivity preserving. Another distinctive feature of our scheme is that it is entropy stable for both the thermodynamic and mathematical entropies. Numerical experiments validate the designed high convergence orders of the scheme and demonstrate its good performances for discontinuous problems.

An efficient second-order cell-centered Lagrangian discontinuous Galerkin method for two-dimensional elastic-plastic flows

An efficient second-order cell-centered Lagrangian discontinuous Galerkin (DG) method for solving two-dimensional (2D) elastic-plastic flows with the hypo-elastic constitutive model and von Mises yield condition is presented. First, starting from the governing equations of conserved quantities in the Euler framework, the integral weak formulation of them in the Lagrangian framework is derived. Next, the DG method is used for spatial discretization of both the weak formulation of conserved quantities and the evolution equation of deviatoric stress tensor. The Taylor basis functions defined in the reference coordinates provide the piecewise polynomial expansion of the variables, including the conserved quantities and the deviatoric stress tensor. The vertex velocities and Cauchy stress tensor on the edges are computed using a nodal solver equipped with a variant of Li's new Harten-Lax-van Leer-contact approximate Riemann solver [Li et al., “An HLLC-type approximate Riemann solver for two-dimensional elastic-perfectly plastic model,” J. Comput. Phys. 448, 110675 (2022)], in which the longitudinal wave velocity in the plastic state is modified. Then the vertex velocities and Cauchy stress tensor on the edges are used to compute numerical fluxes. A second-order total variation diminishing Runge–Kutta scheme is used for time discretization of both the governing equations of conserved quantities and the evolution equation of deviatoric stress tensor. After solving the evolution equation of deviatoric stress tensor, a radial return algorithm is performed at the Gauss points of each element according to the von Mises yield condition. And then the coefficients of the DG expansion for the deviatoric stress tensor on each element are modified by a least squares procedure using the deviatoric stress tensors at these Gauss points. To achieve second-order accuracy, the least squares procedure is used for piecewise linear reconstruction of conserved quantities and the deviatoric stress tensor, and the Barth–Jespersen limiter is used to suppress the nonphysical numerical oscillation near the discontinuities. After that, the coefficients of the DG expansion are modified through L2 projection using the reconstructed polynomials. Finally, a second-order cell-centered Lagrangian DG scheme is established. Several tests demonstrate that the new scheme achieves second-order accuracy with good robustness, and that the DG method of updating the deviatoric stress tensor has comparable accuracy and much higher efficiency with mesh refinement compared with previous works.

Positivity-preserving and entropy-bounded discontinuous Galerkin method for the chemically reacting, compressible Euler equations. Part II: The multidimensional case

In this second part of our two-part paper, we extend to multiple spatial dimensions the one-dimensional, fully conservative, positivity-preserving, and entropy-bounded discontinuous Galerkin scheme developed in the first part for the chemically reacting Euler equations. Our primary objective is to enable robust and accurate solutions to complex reacting-flow problems using the high-order discontinuous Galerkin method without requiring extremely high resolution. Variable thermodynamics and detailed chemistry are considered. Our multidimensional framework can be regarded as a further generalization of similar positivity-preserving and/or entropy-bounded discontinuous Galerkin schemes in the literature. In particular, the proposed formulation is compatible with curved elements of arbitrary shape, a variety of numerical flux functions, general quadrature rules with positive weights, and mixtures of thermally perfect gases. Preservation of pressure equilibrium between adjacent elements, especially crucial in simulations of multicomponent flows, is discussed. Complex detonation waves in two and three dimensions are accurately computed using high-order polynomials. Enforcement of an entropy bound, as opposed to solely the positivity property, is found to significantly improve stability. Mass, total energy, and atomic elements are shown to be discretely conserved.

Optimal control of a semiclassical Boltzmann equation for charge transport in graphene

An ensemble optimal control problem governed by a semiclassical space-homogeneous Boltzmann equation for charge transport in graphene is formulated and analyzed.The control mechanism is an external time-dependent electric field having the purpose of driving the material density of electrons in graphene to follow a desired trajectory. For this purpose, an ensemble cost functional with quadratic H1 costs is considered.Well-posedness of the control problem is proved, and the characterization of optimal controls by an optimality system is discussed. This system is approximated by a discontinuous Galerkin scheme and solved using a nonlinear conjugate gradient method. Results of numerical experiments are presented that demonstrate the ability of the proposed control framework to design control fields.

IMPLICIT EXTENDED DISCONTINUOUS GALERKIN SCHEME FOR SOLVING SINGULARLY PERTURBED BURGERS' EQUATIONS

We present the implicit-modal discontinuous Galerkin scheme for solving the coupled viscous and singularly perturbed Burgers’ equations. This scheme overcomes overshoot and undershoots phenomena in the singularly perturbed Burgers’ equations. We present the stability analysis and obtain suitable ranges for penalty terms and time steps. Also, we gain the constant of trace inequality for the approximate function and its first derivatives based on Legendre basis functions. The numerical results have good agreement with the analytical and available approximate solutions.

A Phase-Space Discontinuous Galerkin Scheme for the Radiative Transfer Equation in Slab Geometry

Abstract We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging.

An Implicit Quadrature-Free Modal Discontinuous Galerkin (DG) Scheme for Shallow Water Equations

An Implicit Quadrature-Free Modal Discontinuous Galerkin (DG) Scheme for Shallow Water Equations

Aircraft Simulations Using the New CFD Software from ONERA, DLR, and Airbus

This paper presents a thorough comparison between RANS simulations performed with the computational fluid dynamics by ONERA, DLR, and Airbus (CODA) new-generation flow solver and reference legacy codes TAU (DLR) and elsA (ONERA) for high-speed cruising NASA Common Research Model (CRM) configurations that were considered in the context of the 5th, 6th, and 7th Drag Prediction Workshops. The solver’s accuracy is assessed with several meshing strategies, including block-structured, hybrid-structured/unstructured, and fully unstructured tetrahedral meshes. This solver features both a cell-centered finite-volume (FV) scheme suited to arbitrary meshes as well as a modern high-order discontinuous Galerkin (DG) scheme. We show that for all cases considered, the FV component recovers an equivalent accuracy compared to elsA (cell-centered FV) on block-structured meshes and TAU (node-centered FV) on hybrid and unstructured meshes. The high-order DG scheme (third-order accurate) is found to significantly enhance the drag prediction on coarse meshes compared to legacy FV methods, both for structured and unstructured meshes.

A discontinuous Galerkin Method based on POD model reduction for Euler equation

<abstract> <p>This paper considers the work of combining the proper orthogonal decomposition (POD) reduced-order method with the discontinuous Galerkin (DG) method to solve three-dimensional time-domain Euler equations. The POD-DG formulation is established by constructing the POD base vector space, based on POD technology one can apply the Galerkin projection of the DG scheme to this dimension reduction space for calculation. Its overall goal is to overcome the disadvantages of high computational cost and memory requirement in the DG algorithm, reduce the degrees of freedom (DOFs) of the calculation model, and save the calculation time while ensuring acceptable accuracy. Numerical experiments verify these advantages of the proposed POD-DG method.</p> </abstract>