Discontinuous element pressure gradient stabilizations for compressible Navier-Stokes equations based on local projections

A pressure gradient discontinuous finite element formulation for the compressible Navier-Stokes equations is derived based on local projections. The resulting finite element formulation is stable and uniquely solvable without requiring a B-B stability condition. An error estimate is Obtained.

Panel Acoustic Contribution Analysis in Automotive Acoustics Using Discontinuous Isogeometric Boundary Element Method

In automotive industries,panel acoustic contribution analysis(PACA)is used to investigate the contributions of the body panels to the acoustic pressure at a certain point of interest.Currently,PACA is implementedmostly by either experiment-based methods or traditional numerical methods.However,these schemes are effort-consuming and inefficient in solving engineering problems,thereby restraining the further development of PACA in automotive acoustics.In this work,we propose a PACA scheme using discontinuous isogeometric boundary element method(IGABEM)to build an easily implementable and efficient method to identify the relative acoustic contributions of each automotive body panel.Discontinuous IGABEMis more accurate and converges faster than continuous BEM and IGABEM in the interior sound pressure evaluation of automotive compartments.In this work,a contribution ratio is defined to estimate the relative acoustic contribution of the structure panels;it can be calculated by reusing the coefficient matrix that has already been generated in the sound pressure evaluation process.The utilization of the parallel technique enables the proposed method to be more efficient than conventional methods;it is validated in two numerical examples,including a car passenger compartment subjected to realistic boundary conditions.A sound pressure response experiment based on a steel box is conducted to verify the accuracy of the interior sound pressure calculation using discontinuous IGABEM.This work is expected to promote the practical process of IGABEM for application in automotive acoustic problems.

A Sub-element Adaptive Shock Capturing Approach for Discontinuous Galerkin Methods

In this paper,a new strategy for a sub-element-based shock capturing for discontinuous Galerkin(DG)approximations is presented.The idea is to interpret a DG element as a col-lection of data and construct a hierarchy of low-to-high-order discretizations on this set of data,including a first-order finite volume scheme up to the full-order DG scheme.The dif-ferent DG discretizations are then blended according to sub-element troubled cell indicators,resulting in a final discretization that adaptively blends from low to high order within a single DG element.The goal is to retain as much high-order accuracy as possible,even in simula-tions with very strong shocks,as,e.g.,presented in the Sedov test.The framework retains the locality of the standard DG scheme and is hence well suited for a combination with adaptive mesh refinement and parallel computing.The numerical tests demonstrate the sub-element adaptive behavior of the new shock capturing approach and its high accuracy.

A multithreaded parallel upwind sweep algorithm for the S_(N) transport equations discretized with discontinuous finite elements

The complex structure and strong heterogeneity of advanced nuclear reactor systems pose challenges for high-fidelity neutron-shielding calculations. Unstructured meshes exhibit strong geometric adaptability and can overcome the deficiencies of conventionally structured meshes in complex geometry modeling. A multithreaded parallel upwind sweep algorithm for S_(N) transport was proposed to achieve a more accurate geometric description and improve the computational efficiency. The spatial variables were discretized using the standard discontinuous Galerkin finite-element method. The angular flux transmission between neighboring meshes was handled using an upwind scheme. In addition, a combination of a mesh transport sweep and angular iterations was realized using a multithreaded parallel technique. The algorithm was implemented in the 2D/3D S_(N) transport code ThorSNIPE, and numerical evaluations were conducted using three typical benchmark problems:IAEA, Kobayashi-3i, and VENUS-3. These numerical results indicate that the multithreaded parallel upwind sweep algorithm can achieve high computational efficiency. ThorSNIPE, with a multithreaded parallel upwind sweep algorithm, has good reliability, stability, and high efficiency, making it suitable for complex shielding calculations.

Adaptive discontinuous finite element quadrature sets over an icosahedron for discrete ordinates method

The discrete ordinates(S N)method requires numerous angular unknowns to achieve the desired accu-racy for shielding calculations involving strong anisotropy.Our objective is to develop an angular adaptive algorithm in the S N method to automatically optimize the angular distribution and minimize angular discretization errors with lower expenses.The proposed method enables linear dis-continuous finite element quadrature sets over an icosahe-dron to vary their quadrature orders in a one-twentieth sphere so that fine resolutions can be applied to the angular domains that are important.An error estimation that operates in conjunction with the spherical harmonics method is developed to determine the locations where more refinement is required.The adaptive quadrature sets are applied to three duct problems,including the Kobayashi benchmarks and the IRI-TUB research reactor,which emphasize the ability of this method to resolve neutron streaming through ducts with voids.The results indicate that the performance of the adaptive method is more effi-cient than that of uniform quadrature sets for duct transport problems.Our adaptive method offers an appropriate placement of angular unknowns to accurately integrate angular fluxes while reducing the computational costs in terms of unknowns and run times.

Mixed time discontinuous space-time finite element method for convection diffusion equations

A mixed time discontinuous space-time finite element scheme for secondorder convection diffusion problems is constructed and analyzed. Order of the equation is lowered by the mixed finite element method. The low order equation is discretized with a space-time finite element method, continuous in space but discontinuous in time. Stability, existence, uniqueness and convergence of the approximate solutions are proved. Numerical results are presented to illustrate efficiency of the proposed method.

Ultraconvergence for averaging discontinuous finite elements and its applications in Hamiltonian system

This paper discusses the k-degree averaging discontinuous finite element solution for the initial value problem of ordinary differential equations. When k is even, the averaging numerical flux （the average of left and right limits for the discontinuous finite element at nodes） has the optimal-order ultraconvergence 2k ＋ 2. For nanlinear Hamiltonian systems （e.g., SchrSdinger equation and Kepler system） with momentum conservation, the discontinuous finite element methods preserve momentum at nodes. These properties are confirmed by numerical experiments.

A Residual Distribution Method Using Discontinuous Elements for the Computation of Possibly Non Smooth Flows

In this paper,we describe a residual distribution(RD)method where,contrarily to“standard”this type schemes,the mesh is not necessarily conformal.It also allows to use discontinuous elements,contrarily to the“standard”case where continuous elements are requested.Moreover,if continuity is forced,the scheme is similar to the standard RD case.Hence,the situation becomes comparable with the Discontinuous Galerkin(DG)method,but it is simpler to implement than DG and has guaranteed L^(∞)bounds.We focus on the second-order case,but the method can be easily generalized to higher degree polynomials.

The discontinuous Petrov-Galerkin method for one-dimensional compressible Euler equations in the Lagrangian coordinate

In this paper, a Petrov-Galerkin scheme named the Runge-Kutta control volume （RKCV） discontinuous finite ele- ment method is constructed to solve the one-dimensional compressible Euler equations in the Lagrangian coordinate. Its advantages include preservation of the local conservation and a high resolution. Compared with the Runge-Kutta discon- tinuous Galerkin （RKDG） method, the RKCV method is easier to implement. Moreover, the advantages of the RKCV and the Lagrangian methods are combined in the new method. Several numerical examples are given to illustrate the accuracy and the reliability of the algorithm.

Discontinuous-Galerkin-Based Analysis of Traffic Flow Model Connected with Multi-Agent Traffic Model

As the number of automobiles continues to increase year after year,the associated problem of traffic congestion has become a serious societal issue.Initiatives to mitigate this problem have considered methods for optimizing traffic volumes in wide-area road networks,and traffic-flow simulation has become a focus of interest as a technique for advance characterization of such strategies.Classes of models commonly used for traffic-flow simulations include microscopic models based on discrete vehicle representations,macroscopic models that describe entire traffic-flow systems in terms of average vehicle densities and velocities,and mesoscopic models and hybrid(or multiscale)models incorporating both microscopic and macroscopic features.Because traffic-flow simulations are designed to model traffic systems under a variety of conditions,their underlyingmodelsmust be capable of rapidly capturing the consequences of minor variations in operating environments.In other words,the computation speed of macroscopic models and the precise representation of microscopic models are needed simultaneously.Thus,in this study we propose a multiscale model that combines a microscopic model—for detailed analysis of subregions containing traffic congestion bottlenecks or other localized phenomena of interest-with a macroscopic model enabling simulation of wide target areas at a modest computational cost.In addition,to ensure analytical stability with robustness in the presence of discontinuities,we discretize our macroscopic model using a discontinuous Galerkin finite element method(DGFEM),while to conjoin microscopic and macroscopic models,we use a generating/absorbing sponge layer,a technique widely used for numerical analysis of long-wavelength phenomena in shallow water,to enable traffic-flow simulations with stable input and output regions.

THE OPTIMAL CONVERGENCE ORDER OF THE DISCONTINUOUS FINITE ELEMENT METHODS FOR FIRST ORDER HYPERBOLIC SYSTEMS

In this paper, a discontinuous finite element method for the positive and symmetric, first-order hyperbolic systems （steady and nonsteady state） is constructed and analyzed by using linear triangle elements, and the O（h^2）-order optimal error estimates are derived under the assumption of strongly regular triangulation and the Ha-regularity for the exact solutions. The convergence analysis is based on some superclose estimates of the interpolation approximation. Finally, we discuss the Maxwell equations in a two-dimensional domain, and numerical experiments are given to validate the theoretical results.

DISCONTINUOUS FINITE ELEMENT METHOD FOR CONVECTION-DIFFUSION EQUATIONS

A discontinuous finite element method for convection-diffusion equations is proposed and analyzed. This scheme is designed to produce an approximate solution which is completely discontinuous. Optimal order of convergence is obtained for model problem. This is the same convergence rate known for the classical methods. [ABSTRACT FROM AUTHOR]

A NEW DIRECT DISCONTINUOUS GALERKIN METHOD WITH SYMMETRIC STRUCTURE FOR NONLINEAR DIFFUSION EQUATIONS

In this paper we continue the study of discontinuous Galerkin finite element methods for nonlinear diffusion equations following the direct discontinuous Galerkin （DDG） meth- ods for diffusion problems [17] and the direct discontinuous Galerkin （DDG） methods for diffusion with interface corrections [18]. We introduce a numerical flux for the test func- tion, and obtain a new direct discontinuous Galerkin method with symmetric structure. Second order derivative jump terms are included in the numerical flux formula and explicit guidelines for choosing the numerical flux are given. The constructed scheme has a sym- metric property and an optimal L2 （L2） error estimate is obtained. Numerical examples are carried out to demonstrate the optimal （k ＋ 1）th order of accuracy for the method with pk polynomial approximations for both linear and nonlinear problems, under one-dimensional and two-dimensional settings.

Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids.In order to obtain a sufficiently stable higher order scheme with respect to the time and space coordinates,we develop a combination of the discontinuous Galerkin finite element(DGFE)method for the space discretization and the backward difference formulae(BDF)for the time discretization.Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step,we employ suitable linearizations of inviscid as well as viscous fluxes which give a linear algebraic problem at each time step.Finally,the resulting BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results are compared with reference data.

MIXED DISCONTINUOUS GALERKIN TIME-STEPPING METHOD FOR LINEAR PARABOLIC OPTIMAL CONTROL PROBLEMS

In this paper, we discuss the mixed discontinuous Galerkin （DG） finite element ap- proximation to linear parabolic optimal control problems. For the state variables and the co-state variables, the discontinuous finite element method is used for the time dis- cretization and the Raviart-Thomas mixed finite element method is used for the space discretization. We do not discretize the space of admissible control but implicitly utilize the relation between co-state and control for the discretization of the control. We de- rive a priori error estimates for the lowest order mixed DG finite element approximation. Moveover, for the element of arbitrary order in space and time, we derive a posteriori L2（O, T; L2（Ω）） error estimates for the scalar functions, assuming that only the underlying mesh is static. Finally, we present an example to confirm the theoretical result on a priori error estimates.

DISCONTINUOUS GALERKIN CALCULATIONS FOR A NONLINEAR PDE MODEL OF DNA TRANSCRIPTION WITH SHORT, TRANSIENT AND FREQUENT PAUSING

A discontinuous Galerkin finite element method is used to approximate solutions to a classical traffic flow PDE. This PDE is used to model the biological process of transcription; the process of transferring genetic information from DNA either to mRNA or to rRNA. The transcription process is punctuated by short, frequent RNAP pauses which are incorporated into the model as traffic lights. These pauses cause a delay in the average transcription process. The DG solution of the nonlinear model is used to calculate the delay and to determine the effect of the pauses on the average transcription time. Numerical error measurements between the DG solution and the true solution （derived by the method of characteristics） are given for a simple model problem. It shows an excellent agreement in a neighborhood away from the shocks as well as C9（Ax） convergence for the delay calculation. Preliminary parameter studies indicate that in a system with multiple pauses both the location and time duration of the pauses can significantly affect the average delay experienced by an RNAP.