Discontinuous Galerkin - Science topic

In this paper, we study the fractional order Rayleigh-Stokes problem, where the time-fractional derivative is considered in the sense of Caputo with order \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsi...

We present a realizability-preserving numerical method for solving a spectral two-moment model to simulate the transport of massless, neutral particles interacting with a steady background material moving with relativistic velocities. The model is obtained as the special relativistic limit of a four-momentum-conservative general relativistic two-mo...

In this work, a numerical method is introduced for the study of nonlinear interactions between free-surface shallow-water flows and a partly immersed floating object. At the continuous level, the fluid’s evolution is modeled with the nonlinear hyperbolic shallow-water equations. The description of the flow beneath the object reduces to an algebraic...

Particle collisions are the primary mechanism of inter-particle momentum and energy exchange for dense particle-laden flow. Accurate approximation of this collision operator in four-way coupled Euler-Lagrange approaches remains challenging due to the associated computational cost. Adopting a deterministic collision model and a hard-sphere approach...

As an effective geophysical tool, ground penetrating radar (GPR) is widely used for environmental and engineering detections. Numerous numerical simulation algorithms have been developed to improve the computational efficiency of GPR simulations, enabling the modeling of complex structures. The discontinuous Galerkin method is a high efficiency num...

We present Sapphire++, an open-source code designed to numerically solve the Vlasov-Fokker-Planck equation for astrophysical applications. Sapphire++ employs a numerical algorithm based on a spherical harmonic expansion of the distribution function, expressing the Vlasov-Fokker-Planck equation as a system of partial differential equations governing...

This work presents a hybrid pressure face-centred finite volume (FCFV) solver to simulate steady-state incompressible Navier-Stokes flows. The method leverages the robustness, in the incompressible limit, of the hybridisable discontinuous Galerkin paradigm for compressible and weakly compressible flows to derive the formulation of a novel, low-orde...

Binary black holes are the most abundant source of gravitational-wave observations. Gravitational-wave observatories in the next decade will require tremendous increases in the accuracy of numerical waveforms modeling binary black holes, compared to today’s state of the art. One approach to achieving the required accuracy is using spectral-type met...

Wall shear stress (WSS) is a crucial hemodynamic quantity extensively studied in cardiovascular research, yet its numerical computation is not straightforward. This work aims to compare WSS results obtained from two different finite element discretizations, quantify the differences between continuous and discontinuous stresses, and introduce a nove...

We conduct the stability analysis of discontinuous Galerkin methods applied to Volterra integral equations in this paper. Stability conditions with respect to both the basic and convolution test equations are derived. Our findings indicate that the methods with orders up to 6 exhibit A$$ A $$‐stability when applied with the basic test equation, whi...

The treatment of material interface and cavitation in compressible flow brings difficulties and challenges for numerical simulation, which is also a research field of great significance. Therefore, we present a discontinuous Galerkin (DG) method to simulate cavitation in multiphase flow by combining the γ-based model and a cutoff cavitation model....

A numerical method ADER-DG with a local DG predictor for solving a DAE system has been developed, which was based on the formulation of numerical methods ADER-DG using a local DG predictor for solving ODE and PDE systems. The basis functions were chosen in the form of Lagrange interpolation polynomials with nodal points at the roots of the right Ra...

A variational formulation based on velocity and stress is developed for linear fluid–structure interaction problems. The well-posedness and energy stability of this formulation are established. A hybridizable discontinuous Galerkin method is employed to discretize the problem. An hp-convergence analysis is performed for the resulting semi-discrete...

In this paper, we propose a class of efficient high order accurate conservative oscillation-eliminating discontinuous Galerkin (OEDG) methods combined with explicit-implicit-null (EIN) time discretizations for solving non-equilibrium three-temperature (3-T) radiation hydrodynamics (RHD) equations. The system of 3-T RHD equations consisting of advec...

This paper is concerned with the numerical solution of third-kind Volterra integral equations with non-smooth kernels. Discontinuous Galerkin methods and two postprocessing techniques are introduced. First, the existence and uniqueness of the numerical solution are established. Then, the convergence of the numerical solutions and the global superco...

For solving hyperbolic conservation laws that arise frequently in computational physics, high order finite volume WENO (FV-WENO) schemes and discontinuous Galerkin (DG) methods are more popular because of their applicability to any monotone fluxes and easily handle complicated geometries and boundary conditions. However, when there are smaller scal...

In this paper, we introduce and solve a novel model that integrates confined flow within a dual porosity poroelastic medium with free flow in conduits. The model is structured around three distinct but interconnected regions: the matrix, micro-fractures, and conduits. Fluid flow within the dual porosity poroelastic medium is described by a dual-por...

The discontinuous Galerkin (DG) method has been widely considered in recent years to develop scalable flow solvers for its ability to handle discontinuities, such as shocks and detonations, with greater accuracy and high arithmetic intensity. However, its scalability is severely affected by communication bottlenecks that arise from data movement an...

This work presents GALÆXI as a novel, energy-efficient flow solver for the simulation of compressible flows on unstructured hexahedral meshes leveraging the parallel computing power of modern Graphics Processing Units (GPUs). GALÆXI implements the high-order Discontinuous Galerkin Spectral Element Method (DGSEM) using shock capturing with a finite-...

High-order methods have recently been shown to be an effective tool for high-fidelity flow computations like direct numerical simulations and large eddy simulations due to their strong balance between accuracy and computational cost. In this work, a high-order discontinuous Galerkin spectral element method (DGSEM) is developed to solve the chemical...

This work aims at presenting a discontinuous Galerkin (DG) formulation employing a spectral basis for two important models employed in cardiac electrophysiology, namely the monodomain and bidomain models. The use of DG methods is motivated by the characteristic of the mathematical solution of such equations which often corresponds to a highly steep...

This article examines the convergence of the two-step backward difference formula (BDF2)-discontinuous Galerkin (DG) scheme in the context of a nonlocal reaction–diffusion equation. Our methodology integrates BDF2 for temporal discretization with a DG finite element method for spatial discretization, enabling a comprehensive analysis of the nonloca...

In this paper we consider time-dependent PDEs discretized by a special class of Physics Informed Neural Networks whose design is based on the framework of Runge--Kutta and related time-Galerkin discretizations. The primary motivation for using such methods is that alternative time-discrete schemes not only enable higher-order approximations but als...

A new numerical method, which is based on the coupling of adaptive mesh technique, level set (LS) method, square-root-conformation representation (SRCR) approach, and discontinuous Galerkin (DG) method within the dual splitting framework, is developed for viscoelastic two-phase flow problems. This combination has been more effective than expected....

We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate KdV solitons, as well as other solitary and periodic solutions that are related to higher-order water wave mod...

We present a novel spatial discretization for the anisotropic heat conduction equation, aimed at improved accuracy at the high levels of anisotropy seen in a magnetized plasma, for example, for magnetic confinement fusion. The new discretization is based on a mixed formulation, introducing a form of the directional derivative along the magnetic fie...

We present a novel technique for imposing non-linear entropy conservative and entropy stable wall boundary conditions for the resistive magnetohydrodynamic equations in the presence of an adiabatic wall or a wall with a prescribed heat entropy flow, addressing three scenarios: electrically insulating walls, thin walls with finite conductivity, and...

We present a quantitative assessment of the impact of high-order mappings on the simulation of flows over complex orography. Curved boundaries were not used in early numerical methods, whereas they are employed to an increasing extent in state of the art computational fluid dynamics codes, in combination with high-order methods, such as the Finite...

In this paper, we compare the intrusive proper orthogonal decomposition (POD) with Galerkin projection and the data-driven dynamic mode decomposition (DMD), for Hes-ton's option pricing model. The full order model is obtained by discontinuous Galerkin discretization in space and backward Euler in time. Numerical results for butterfly spread, Europe...

In the first part of this paper, uniqueness of strong solution is established for the Vlasov-unsteady Stokes problem in 3D. The second part deals with a semi discrete scheme, which is based on the coupling of discontinuous Galerkin approximations for the Vlasov and the Stokes equations for the 2D problem. The proposed method is both mass and moment...

We provide a posteriori error estimates for a discontinuous Galerkin scheme for the parabolic-elliptic Keller-Segel system in 2 or 3 space dimensions. The estimates are conditional in the sense that an a posteriori computable quantity needs to be small enough—which can be ensured by mesh refinement—and optimal in the sense that the error estimator...

Clamp-on ultrasonic flowmeters suffer from crosstalk-i.e., measurement errors due to the interference of signals generated in solid regions and solid-fluid interfaces with the required signal from the fluid. Although several approaches have been proposed to alleviate crosstalk, they only work in specific ranges of flow rates and pipe diameters, and...

We consider the discretization of a class of nonlinear parabolic equations by discontinuous Galerkin time-stepping methods and establish a priori as well as conditional a posteriori error estimates. Our approach is motivated by the error analysis in [9] for Runge-Kutta methods for nonlinear parabolic equations; in analogy to [9], the proofs are bas...

This work introduces finite element methods for a class of elliptic fully nonlinear partial differential equations. They are based on a minimal residual principle that builds upon the Alexandrov--Bakelman--Pucci estimate. Under rather general structural assumptions on the operator, convergence of $C^1$ conforming and discontinuous Galerkin methods...

The separation of fast (barotropic) and slow (baroclinic) motions into subsystems through barotropic-baroclinic splitting has been widely adopted in layered ocean circulation models. To date, the majority of models use finite difference or finite volume methods alongside this splitting technique. In this paper, we present an extension of the work i...

This work addresses the challenge of shock capturing in numerical simulations of hyperbolic conservation laws, focusing on the discontinuous Galerkin (DG) method. It proposes a novel modal filtering approach based on physical criteria to detect elements near discontinuities and tune modal damping in the solutions. The minimum entropy principle comb...

The computation of electric field in transcranial magnetic stimulation (TMS) is essentially a problem of gradient calculation for thin layers. This paper introduces a hybrid-order hybridizable discontinuous Galerkin finite element method (HDG-FEM) and systematically demonstrates its superiority in TMS computations. The discrete format of HDG-FEM em...

This paper presents a unifying framework for Trefftz-like methods, which allows the analysis and construction of discretization methods based on the decomposition into, and coupling of, local and global problems. We apply the framework to provide a comprehensive error analysis for the Embedded Trefftz discontinuous Galerkin method, for a wide range...

The aim of this paper is to introduce, analyse and test in practice a new mathematical model describing the interplay between biological tissue atrophy driven by pathogen diffusion, with applications to neurodegenerative disorders. This study introduces a novel mathematical and computational model comprising a Fisher-Kolmogorov equation for species...

Mathematical models of protein-protein dynamics, such as the heterodimer model, play a crucial role in understanding many physical phenomena, e.g., the progression of some neu-rodegenerative diseases. This model is a system of two semilinear parabolic partial differential equations describing the evolution and mutual interaction of biological speci...

This study computationally examined the Richtmyer–Meshkov instability (RMI) evolution in a helium backward-triangular bubble immersed in monatomic argon, diatomic nitrogen, and polyatomic methane under planar shock wave interactions. Using high-fidelity numerical simulations based on the compressible Navier–Fourier equations based on the Boltzmann–...

In this paper we consider the optimal control problems governed by the gradient systems for the Ginzburg-Landau free energy where denotes a potential function and ε is the diffusivity. One example of gradient systems are the Schlögl equation arising in chemical waves with a quartic potential function F(y). Gradient systems are characterized by ener...

We present and analyze a discontinuous Galerkin method for the numerical modeling of a Kelvin-Voigt thermo/poro-viscoelastic problem. We present the derivation of the model, and we develop a stability analysis in the continuous setting that holds both for the full inertial and quasi-static problems and that is robust with respect to most of the phy...

This paper proposes a discussion of the direct discontinuous Galerkin (DDG) methods coupled with explicit-implicit-null time discretizations (EIN) for solving the nonlinear diffusion equation u t = (a(u)u x) x. The basic idea of the EIN method is to add and subtract two equal constant coefficient terms a 1 u xx (a 1 = a 0 × max u a(u)) on the right...

The modified transmission eigenvalue (mTE) problem, introduced by Cogar et al. (Inverse Probl. 33, 125002, 2017) and Audibert et al. (Inverse Probl. 33, 125011, 2017), plays a fundamental role in the theoretical and numerical investigations of the inverse medium problem. In this paper, we study the discontinuous Galerkin discretization for the mTE...

In two dimensions, we propose and analyse an iterative a posteriori error indicator for the discontinuous Galerkin finite element approximations of the Stokes equations under boundary conditions of friction type. Two sources of error are identified here, namely; the discretisation error and the linearization error. Under a smallness assumption on d...

Neutron interactions in a fusion power plant play a pivotal role in determining critical design parameters such as coil-plasma distance and breeding blanket composition. Fast predictive neutronic capabilities are therefore crucial for an efficient design process. For this purpose, we have developed a new deterministic neutronics method, capable of...

This paper deals with a fully discrete numerical scheme for the incompressible Chemotaxis(Keller-Segel)-Navier-Stokes system. Based on a discontinuous Galerkin finite element scheme in the spatial directions, a semi-implicit first-order finite difference method in the temporal direction is applied to derive a completely discrete scheme. With the he...

Based on the Jacobi polynomial expansion, an arbitrary high-order Discontinuous Galerkin solver for compressible flows on unstructured meshes is proposed in the present work. First, we construct orthogonal polynomials for 2D and 3D isoparametric elements using the 1D Jacobi polynomials. We perform modal expansions of the state variables using the o...

In this article, a hybridizable discontinuous Galerkin (HDG) method is proposed and analyzed for the Klein-Gordon equation with local Lipschitz-type non-linearity. {\it A priori} error estimates are derived, and it is proved that approximations of the flux and the displacement converge with order $O(h^{k+1}),$ where $h$ is the discretizing paramete...

We propose and analyze a space-time Local Discontinuous Galerkin method for the approximation of the solution to parabolic problems. The method allows for very general discrete spaces and prismatic space-time meshes. Existence and uniqueness of a discrete solution are shown by means of an inf-sup condition, whose proof does not rely on polynomial i...

This paper presents a novel parallel-GPU discontinuous Galerkin time domain (DGTD) method with a third-order local time stepping (LTS) scheme for the solution of multi-scale electromagnetic problems. The parallel-GPU implementations were developed based on NVIDIA’s recommendations to guarantee the optimal GPU performance, and an LTS scheme based on...

This paper introduces and analyzes a staggered discontinuous Galerkin (DG) method for quasi-Newtonian Stokes flow problems on polytopal meshes. The method introduces the flux and tensor gradient of the velocity as additional unknowns and eliminates the pressure variable via the incompressibility condition. Thanks to the subtle construction of the f...

This paper introduces a novel numerical integration scheme tailored for polytopic domains, circumventing the need for sub-tessellation or sub-tetrahedralization. Our method involves defining integration points on a Cartesian bounding box surrounding the polytopic domain and computing integration weights through moment matching with analytically com...

Adaptive mesh refinement (AMR) technology and high-order methods are important means to improve the quality of simulation results and have been hotspots in the computational fluid dynamics community. In this paper, high-order discontinuous Galerkin (DG) and direct DG (DDG) finite element methods are developed based on a parallel adaptive Cartesian...

In this article, we develop a linear, fully discrete numerical scheme for solving the fourth-order Cahn-Hilliard equation with the general type nonlinear potential, where spatial discretization is based on the hybridizable discontinuous Galerkin method, and the time discretization is based on the recently developed Invariant Energy Quadratization a...

Heterogeneous systems, such as composites, are common in many areas of engineering. In recent years, there has been interest in using mesoscale modeling to gain insight into the mechanical behavior and failure of composites. However, when using the conventional finite element method, it can be very difficult to create high-quality body-fitted meshe...

In this work we present an a priori error analysis for solving the unsteady advection equation on cut cell meshes along a straight ramp in two dimensions. The space discretization uses a lowest order upwind-type discontinuous Galerkin scheme involving a \textit{Domain of Dependence} (DoD) stabilization to correct the update in the neighborhood of s...

In Part I, we optoelectronically optimized a thin-film solar cell with a graded-bandgap CZTSSe photon-absorbing layer and a periodically corrugated backreflector, using the hybridizable discontinuous Galerkin (HDG) scheme to solve the drift-diffusion equations. The efficiency increase due to periodic corrugation was minimal, but significant improve...

A multi-component non-isothermal lumped kinetic model (LKM) of fixed-bed liquid chromatographic reactor is introduced and numerically approximated. The model incorporates nonlinear Bi-Langmuir and Toth adsorption isotherms, reaction kinetics, axial and temporal variations of concentrations, and enthalpies of reaction and adsorption. The resulting m...

A transient circularly polarized excitation and its implementation in a generalized dispersive material model based discontinuous Galerkin time-domain solver are proposed for spectral analysis of chiral nanophotonic structures. The expression of a circularly polarized pulse with a certain bandwidth, which is real-valued and enables multi-physics an...

In this paper, the spatial local discontinuous Galerkin (LDG) approximation coupled with the temporal multistep implicit-explicit (IMEX) evolution for the micropolar Navier–Stokes equations (MNSE) is adopted to construct an efficient fully discrete method. First, the multistep IMEX-LDG methods are constructed up to the third order, which have small...

Moist thermodynamics is a fundamental driver of atmospheric dynamics across all scales, making accurate modeling of these processes essential for reliable weather forecasts and climate change projections. However, atmospheric models often make a variety of inconsistent approximations in representing moist thermodynamics. These inconsistencies can i...