Discontinuous Galerkin Method Research Articles

Convergence of an adaptive C0-interior penalty Galerkin method for the biharmonic problem

Abstract We develop a basic convergence analysis for an adaptive ${C}^{0}\mathsf{IPG}$ method for the biharmonic problem that provides convergence without rates for all practically relevant marking strategies and all penalty parameters assuring coercivity of the method. The analysis hinges on embedding properties of (broken) Sobolev and BV spaces, and the construction of a suitable limit space. In contrast to the previous convergence result of adaptive discontinuous Galerkin methods for elliptic PDEs, here we have to deal with the fact that the Lagrange finite element spaces may possibly contain no proper $C^{1}$-conforming subspace. This prevents from a straight forward generalization and requires the development of some new key technical tools.

The jump filter in the discontinuous Galerkin method for hyperbolic conservation laws

When simulating hyperbolic conservation laws with discontinuous solutions, high-order linear numerical schemes often produce undesirable spurious oscillations. In this paper, we propose a jump filter within the discontinuous Galerkin (DG) method to mitigate these oscillations. This filter operates locally based on jump information at cell interfaces, targeting high-order polynomial modes within each cell. Besides its localized nature, our proposed filter preserves key attributes of the DG method, including conservation, L2 stability, and high-order accuracy. We also explore its compatibility with other damping techniques, and demonstrate its seamless integration into a hybrid limiter. In scenarios featuring strong shock waves, this hybrid approach, incorporating this jump filter as the low-order limiter, effectively suppresses numerical oscillations while exhibiting low numerical dissipation. Additionally, the proposed jump filter maintains the compactness of the DG scheme, which greatly aids in efficient parallel computing. Moreover, it boasts an impressively low computational cost, given that no characteristic decomposition is required and all computations are confined to physical space. Numerical experiments validate the effectiveness and performance of our proposed scheme, confirming its accuracy and shock-capturing capabilities.

On a posteriori error estimation for Runge–Kutta discontinuous Galerkin methods for linear hyperbolic problems

AbstractA posteriori bounds for the error measured in various norms for a standard second‐order explicit‐in‐time Runge–Kutta discontinuous Galerkin (RKDG) discretization of a one‐dimensional (in space) linear transport problem are derived. The proof is based on a novel space‐time polynomial reconstruction, hinging on high‐order temporal reconstructions for continuous and discontinuous Galerkin time‐stepping methods. Of particular interest is the question of error estimation under dynamic mesh modification. The theoretical findings are tested by numerical experiments.

Discontinuous Galerkin Methods for Nonlinear Parabolic Delay-Equations of Nonmonotone Type

Discontinuous Galerkin Methods for Nonlinear Parabolic Delay-Equations of Nonmonotone Type

Robust 3D multi‐material hydrodynamics using discontinuous Galerkin methods

AbstractA high‐order discontinuous Galerkin (DG) method is presented for nonequilibrium multi‐material () flow with sharp interfaces. Material interfaces are reconstructed using the algebraic THINC approach, resulting in a sharp interface resolution. The system assumes stiff velocity relaxation and pressure nonequilibrium. The presented DG method uses Dubiner's orthogonal basis functions on tetrahedral elements. This results in a unique combination of sharp multimaterial interfaces and high‐order accurate solutions in smooth single‐material regions. A novel shock indicator based on the interface conservation condition is introduced to mark regions with discontinuities. Slope limiting techniques are applied only in these regions so that nonphysical oscillations are eliminated while maintaining high‐order accuracy in smooth regions. A local projection is applied on the limited solution to ensure discrete closure law preservation. The effectiveness of this novel limiting strategy is demonstrated for complex three‐dimensional multi‐material problems, where robustness of the method is critical. The presented numerical problems demonstrate that more accurate and efficient multi‐material solutions can be obtained by the DG method, as compared to second‐order finite volume methods.

Accelerated iterative DG finite element solvers for large-scale time-harmonic acoustic problems

Finite element methods are widely used to solve time-harmonic wave propagation problems, but solving large cases can be extremely difficult even with the computational power of parallel computers. In this work, the linear system resulting from the finite element discretization is solved with iterative solution methods, which are efficient in parallel but can require a large number of iterations. In standard discontinuous Galerkin (DG) methods, the numerical solution is discontinuous at the interfaces between the elements. In hybridizable DG methods, additional unknowns are introduced at the interfaces between the finite elements, and the physical unknowns are eliminated from the global system, resulting in a hybridized system. We have recently proposed a new strategy, called CHDG, where the additional unknowns correspond to transmission variables, whereas in the standard approach they are numerical fluxes. This strategy improves the properties of the hybridized system for faster iterative solution procedures. In this talk, we present and study a 3D CHDG implementation with nodal finite element basis functions. The resulting scheme has properties amenable to efficient parallel computing. Numerical results are presented to validate the method, and preliminary 3D computational results are proposed.

Optimizing a small room for critical listening with Treble

This work presents a detailed validation of new software Treble launched in 2023 that hybridizes the discontinuous Galerkin method and pressure-based geometrical acoustics methods. We investigated a small studio for voice productions both through physical measurements and treble's room acoustic simulations. By employing a dense measurement grid with a spacing of 25 cm at the ear level, a comprehensive validation of the room acoustic simulation was conducted in terms of room impulse responses, transfer functions, ISO 3382 acoustic parameters and sound pressure level distribution. Treble's simulation successfully predict the room modes, ISO 3382 acoustic parameters as well as the sound level distribution. The study demonstrates the impact of speaker positions, particularly regarding their interaction with boundary surfaces and how this can be planned with the simulation.

An open-source time-domain wave-based room acoustic software in Python based on the nodal discontinuous Galerkin method

In this work, we introduce an open-source implementation of a time-domain wave-based room acoustic modeling software package, named DG_RoomAcoustics. In this software, the linear acoustic equations are spatially discretized by the nodal discontinuous Galerkin method, and are integrated in time by either the explicit Runge-Kutta or the arbitrary high-order derivatives (ADER) integration schemes. Following the principles of object-oriented programming paradigm, the software is structured to ensure generic applicability and to facilitate future extensions with additional functionalities (e.g., different time integration schemes, boundary conditions). A comprehensive presentation of the physical and numerical aspects of the problem is provided. A detailed exposition of the code structure and components is presented, all of which are released under an open-source license, fostering community feedback and collaborative contributions for ongoing improvements. A brief overview of the current capabilities of the software is introduced. Future work regarding possible functionality extensions and performance optimizing are discussed.

Integrating angular and domain decomposition with space-angle discontinuous Galerkin methods in 2D radiative transfer

A space-angle discontinuous Galerkin (saDG) method is used to solve the steady-state radiative transfer equation (RTE) for 2D problems involving absorption, emission, and scattering for a semitransparent medium. This approach discretizes both spatial and angular domains. Parallel computing is based on angular decomposition (AD), and domain decomposition (DD) techniques. The DD technique directly solves the entire domain using the MUMPS library, whereas the AD technique results in an iterative approach for scattering media. This study proposes a novel hybrid AD-DD method, combining the best aspects of both techniques. Numerical results investigate the scalability, performance, and efficiency of AD and DD techniques. It is shown that a hybrid AD-DD technique is superior to these individual techniques by taking advantage of their strengths. Numerical methods demonstrate the applicability of the method of the best combination of hybrid AD-DD to 2D scattering gray media with complex geometries or enclosures with circular and square obstacles.

A priori and a posteriori error analysis of a mixed DG method for the three-field quasi-Newtonian Stokes flow

Abstract In this paper we propose and analyse a new mixed-type DG method for the three-field quasi-Newtonian Stokes flow. The scheme is based on the introduction of the stress and strain tensor as further unknowns as well as the elimination of the pressure variable by means of the incompressibility constraint. As such, the resulting system involves three unknowns: the stress, the strain tensor and the velocity. All these three unknowns are approximated using discontinuous piecewise polynomials, which offers flexibility for enforcing the symmetry of the stress and the strain tensor. The unique solvability and a comprehensive convergence error analysis for all the variables are performed. All the variables are proved to converge optimally. Adaptive mesh refinement guided by a posteriori error estimator is computationally efficient, especially for problems involving singularity. In line of this mechanism we derive a residual-type a posteriori error estimator, which constitutes the second main contribution of the paper. In particular, we employ the elliptic reconstruction in conjunction with the Helmholtz decomposition to derive the a posteriori error estimator, which avoids using the averaging operator. Several numerical experiments are carried out to verify the theoretical findings.

Efficiency high-order discontinuous Galerkin method for low speeds flows with time-derivative preconditioning

A local preconditioning high-order discontinuous Galerkin (DG) method is introduced to enhance the efficiency and accuracy of the density-based approach for low-speed flows. The method is based on the time-derivative preconditioning technique typically employed in finite volume methods. Traditional approximation methods that only precondition low-order derivatives lead to incompatibility of the equations. To extend the preconditioning technique to the DG method, we employ an analytic preconditioning mass matrix derived from the DG formulation of the preconditioned Navier–Stokes equations. Modified numerical flux functions and a self-adaptive local velocity truncation parameter are employed for DG systems with preconditioning. We demonstrate the efficiency and accuracy of the preconditioned DG method using explicit and implicit schemes for steady flows and a dual-time scheme for unsteady flows. The method has been implemented and used to compute a variety of flow problems, including inviscid and laminar flows, utilizing various degrees of polynomial approximations. Computations with and without preconditioning are performed to analyze the influence of spatial discretization on the accuracy and convergence of the DG solutions at low Mach numbers. Numerical results underscore the enhanced accuracy and convergence properties of the proposed method for low-speed flows, indicating significant performance improvements over traditional methods.

Detective generalized multiscale hybridizable discontinuous Galerkin(GMsHDG) method for porous media

The Detective Generalized Multiscale Hybridizable Discontinuous Galerkin (Detective GMsHDG) method aims to further reduce the computational cost of the GMsHDG method. The GMsHDG method itself reduces the computational cost of the HDG method by employing an upscaled structure on a two-grid mesh. Given a PDE within a specified domain, we subdivide the domain into polygonal subdomains and transforms a HDG problem into globular and local problems. Globular problem concerns whether the solutions on smaller domains glue well to form a globular solution. The process involves generation of multiscale spaces, which is a vector space of functions defined on edges of the polygonal regions. A naive approximation by polynomials fails, especially in porous media, necessitating the generation of problem-specific spaces. The Detective GMsHDG method improves this process by replacing the generation of the multiscale space with the detective algorithm. The Detective GMsHDG method has two stages. First is called an offline stage. During the offline stage, we construct a detective function which, given a permeability distribution, it gives a multiscale space. Later stage is called the offline stage where, given the multiscale space, we use GMsHDG method to solve a given PDE numerically. We show numerical results to argue the liability of the solution using the detective GMsHDG method.

Local subcell monolithic DG/FV convex property preserving scheme on unstructured grids and entropy consideration

This article aims at presenting a new local subcell monolithic Discontinuous-Galerkin/Finite-Volume (DG/FV) convex property preserving scheme solving system of conservation laws on 2D unstructured grids. This is known that DG method needs some sort of nonlinear limiting to avoid spurious oscillations or nonlinear instabilities which may lead to the crash of the code. The main idea motivating the present work is to improve the robustness of DG schemes, while preserving as much as possible its high accuracy and very precise subcell resolution. To do so, a convex blending of high-order DG and first-order FV schemes will be locally performed, at the subcell scale, where it is needed. To this end, by means of the theory developed in [58,59], we first recall that it is possible to rewrite DG scheme as a subcell FV method, defined on a subgrid, provided with some specific numerical fluxes referred to as DG reconstructed fluxes. Then, the subcell monolithic DG/FV method will be defined as follows: to each face of each subcell we will assign two fluxes, a 1st-order FV one and a high-order reconstructed one, that in the end will be blended in a convex way. The goal is then to determine, through analysis, optimal blending coefficients to achieve the desired properties. Numerical results on various type problems will be presented to assess the very good performance of the design method.A particular emphasis will be put on entropy consideration. By means of this subcell monolithic framework, we will attempt to address the following questions: is this possible through this monolithic framework to ensure any entropy stability? What do we mean by entropy stability? What is the cost of such constraints? Is this absolutely needed while aiming for high-order accuracy?

A High-Order Well-Balanced Discontinuous Galerkin Method for Hyperbolic Balance Laws Based on the Gauss-Lobatto Quadrature Rules

In this paper, we develop a high-order well-balanced discontinuous Galerkin method for hyperbolic balance laws based on the Gauss-Lobatto quadrature rules. Important applications of the method include preserving the non-hydrostatic equilibria of shallow water equations with non-flat bottom topography and Euler equations in gravitational fields. The well-balanced property is achieved through two essential components. First, the source term is reformulated in a flux-gradient form in the local reference equilibrium state to mimic the true flux gradient in the balance laws. Consequently, the source term integral is discretized using the same approach as the flux integral at Gauss-Lobatto quadrature points, ensuring that the source term is exactly balanced by the flux in equilibrium states. Our method differs from existing well-balanced DG methods for shallow water equations with non-hydrostatic equilibria, particularly in the aspect that it does not require the decomposition of the source term integral. The effectiveness of our method is demonstrated through ample numerical tests.

A stable decoupled perfectly matched layer for the 3D wave equation using the nodal discontinuous Galerkin method

In outdoor acoustics, the calculations of sound propagating in air can be computationally heavy if the domain is chosen large enough for the sound waves to decay. The computational cost is lowered by strategically truncating the computational domain with an efficient boundary treatment. One commonly used boundary treatment is the perfectly matched layer (PML), which dampens outgoing waves without polluting the computed solution in the inner domain. The purpose of this study is to propose and assess a new perfectly matched layer formulation for the 3D acoustic wave equation, using the nodal discontinuous Galerkin finite element method. The formulation is based on an efficient PML formulation that can be decoupled to further increase the computational efficiency and guarantee stability without sacrificing accuracy. This decoupled PML formulation is demonstrated to be long-time stable, and an optimization procedure for the damping functions is proposed to enhance the performance of the formulation.

An Oscillation-Free Discontinuous Galerkin Method for a Nonlinear Stochastic Convection-Dominated Diffusion Problem and Its Error Analysis

An Oscillation-Free Discontinuous Galerkin Method for a Nonlinear Stochastic Convection-Dominated Diffusion Problem and Its Error Analysis

A new high-order RKDG method based on the TENO-THINC scheme for shock-capturing

In recent years, Runge-Kutta Discontinuous Galerkin (RKDG) methods have gained substantial attention in solving hyperbolic conservation laws, attributed to their high-order accuracy and adaptability to unstructured meshes. However, standard RKDG methods cannot capture discontinuities without oscillation unless they are supplemented with troubled cell indicators and limiters. Existing indicators, such as the total variation bounded (TVB) minmod indicator and the KXRCF indicator, typically depend on a critical parameter tied to the equation's solution, necessitating tuning for different cases. In terms of limiters, the popular limiters even fail to guarantee the high-order property in the smooth regions. The advent of Weighted Essentially Non-Oscillatory (WENO) schemes prompts the implementation of WENO limiters for RKDG methods, preserving high-order properties. However, WENO-family schemes exhibit significant numerical dissipation, potentially smearing small-scale flow structures. In this work, the Targeted Essentially Non-Oscillatory (TENO) indicator [1] is utilized, which leverages the nonlinear weighting strategy of the TENO scheme to separate high-wavenumber physical fluctuations and genuine discontinuities from smooth regions with a unified set of parameters. For troubled cells, a novel limiter is proposed for structured meshes, which combines a TENO scheme for resolving high-wavenumber physical fluctuations and a novel non-polynomial Tangent of Hyperbola for the INterface Capturing (THINC) scheme for resolving genuine discontinuities with extremely low numerical dissipation. Furthermore, the shifting between the TENO and THINC schemes is based on a new boundary variation diminishing (BVD) strategy, which only relies on compact neighborhoods and is significantly simpler than its predecessors. Meanwhile, a new strategy is proposed to ensure the consistency of the new limiter applied for 1D and 2D cases. A set of 1D and 2D benchmark cases including strong shockwaves and a broad range of flow length scales is simulated with uniform meshes for 1D cases and structured quadrilateral meshes for 2D cases to demonstrate the performance of the new numerical scheme. The indicator does not activate any limiters in the accuracy test cases to ensure the high-order property of the whole numerical scheme. Other cases with discontinuities demonstrate the low-dissipation properties of the new limiter in comparison to the simple WENO limiter.

An adaptive hybridizable discontinuous Galerkin method for Darcy–Forchheimer flow in fractured porous media

In this paper, we consider an adaptive hybridizable discontinuous Galerkin (HDG) method based on the discrete fracture model for approximation of a single-phase flow in fractured porous media. We are interested in the case that the flow rate in fracture is large enough to justify the use of Forchheimer’s law for modeling flow within the fracture, while Darcy’s law is applied to the surrounding matrix. The HDG method could be designed to simulate the flow in porous media with reduced fractures which consist of many straight lines or planes. More specifically, we use piecewise polynomials of degree [Formula: see text] to approximate the velocity and pressure in fracture and surrounding porous media. The existence and uniqueness of discrete solutions are proved by the Brouwer fixed point theorem, and an efficient and reliable a posteriori error estimator is obtained with respect to an energy norm. Moreover, the HDG scheme, the existence and uniqueness of discrete solutions, and the a posteriori error estimates are also extended to the problem with non-planar, embedded, and intersecting fractures. Finally, several numerical examples are provided to validate the performance of the obtained a posteriori error estimator.

Uniformly high-order bound-preserving OEDG schemes for two-phase flows

This paper proposes and rigorously analyzes novel high-order bound-preserving oscillation-eliminating discontinuous Galerkin (BP-OEDG) schemes with the Harten–Lax–van Leer (HLL) numerical flux for the Kapila five-equation two-phase flow model. The evolution equation for the volume fraction is formulated as a conservative advection equation with an additional non-conservative source term. Utilizing a technical splitting approach, we design a uniformly high-order discretization for this source term, incorporating an “upwind” discretization of the non-conservative product at cell interfaces based on the HLL wave speeds. This method inherently ensures the symmetry of the volume fractions, which is crucial for establishing the bound-preserving property. The positivity of partial densities and internal energy is rigorously proven using the geometric quasilinearization (GQL) approach, which transforms the nonlinear pressure positivity constraint into equivalent linear constraints. To suppress potential spurious oscillations, we incorporate a scale-invariant and linearity-invariant oscillation elimination (OE) procedure that damps the DG modal coefficients after each Runge–Kutta stage, as proposed in [M. Peng, Z. Sun and K. Wu, OEDG: Oscillation-eliminating discontinuous Galerkin method for hyperbolic conservation laws, Math. Comput. (2024), https://doi.org/10.1090/mcom/3998]. This OE procedure, acting as a post-processing filter based on the jumps of the DG solution at cell interfaces, is easy to implement, maintains the Abgrall equilibrium condition around an isolated material interface, and preserves the high-order accuracy of the DG schemes. The effectiveness and robustness of the proposed high-order BP-OEDG schemes are demonstrated through several benchmark numerical experiments.

Generalized Integral Method for Solving the Neutron Transport Equation with the Hybridized Discontinuous Galerkin Method

Accurate modeling of the neutron transport equation (NTE) with anisotropic scattering is crucial to the understanding of neutron interactions within various mediums. The primary challenges in this domain are (1) the considerable computational resources demanded by anisotropic calculations and (2) the numerical instabilities that arise due to the transport correction approximation. This study introduces a novel, generalized integral method based on the hybridized discontinuous Galerkin framework for solving the second-order NTE with anisotropic scattering. This method employs a spherical harmonics expansion to define the partial current at the mesh interface and applies an angular integral approach to the flux treatment within the mesh. This dual approach facilitates an efficient computational process while preserving accuracy. The integral method has been validated through comparisons with the standard discrete ordinates method (SN) using two eigenvalue problems. The integral method showcases several significant improvements over the traditional SN method. First, it repositions the P0 scattering sources during the formulation process, effectively circumventing the convergence issues associated with transport correction. Second, this strategic repositioning substantially enhances the convergence rates of iterative calculations. Last, a standout feature of the integral method is its capability to perform angular integrals during assembling matrices, successfully reducing the floating-point operations for local flux retrieval and eliminating the ray effect.