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Article
Publication date: 11 May 2010

Adrien Catella, Victorita Dolean and Stéphane Lanteri

The purpose of this paper is to develop a time implicit discontinuous Galerkin method for the simulation of two‐dimensional time‐domain electromagnetic wave propagation on…

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Abstract

Purpose

The purpose of this paper is to develop a time implicit discontinuous Galerkin method for the simulation of two‐dimensional time‐domain electromagnetic wave propagation on non‐uniform triangular meshes.

Design/methodology/approach

The proposed method combines an arbitrary high‐order discontinuous Galerkin method for the discretization in space designed on triangular meshes, with a second‐order Cranck‐Nicolson scheme for time integration. At each time step, a multifrontal sparse LU method is used for solving the linear system resulting from the discretization of the TE Maxwell equations.

Findings

Despite the computational overhead of the solution of a linear system at each time step, the resulting implicit discontinuous Galerkin time‐domain method allows for a noticeable reduction of the computing time as compared to its explicit counterpart based on a leap‐frog time integration scheme.

Research limitations/implications

The proposed method is useful if the underlying mesh is non‐uniform or locally refined such as when dealing with complex geometric features or with heterogeneous propagation media.

Practical implications

The paper is a first step towards the development of an efficient discontinuous Galerkin method for the simulation of three‐dimensional time‐domain electromagnetic wave propagation on non‐uniform tetrahedral meshes. It yields first insights of the capabilities of implicit time stepping through a detailed numerical assessment of accuracy properties and computational performances.

Originality/value

In the field of high‐frequency computational electromagnetism, the use of implicit time stepping has so far been limited to Cartesian meshes in conjunction with the finite difference time‐domain (FDTD) method (e.g. the alternating direction implicit FDTD method). The paper is the first attempt to combine implicit time stepping with a discontinuous Galerkin discretization method designed on simplex meshes.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 29 no. 3
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 3 May 2013

Leilei Wei, Xindong Zhang and Yinnian He

The purpose of this paper is to develop a fully discrete local discontinuous Galerkin (LDG) finite element method for solving a time‐fractional advection‐diffusion equation.

Abstract

Purpose

The purpose of this paper is to develop a fully discrete local discontinuous Galerkin (LDG) finite element method for solving a time‐fractional advection‐diffusion equation.

Design/methodology/approach

The method is based on a finite difference scheme in time and local discontinuous Galerkin methods in space.

Findings

By choosing the numerical fluxes carefully the authors' scheme is proved to be unconditionally stable and gets L2 error estimates of O(hk+1+(Δt)2+(Δt)α/2hk+(1/2)). Finally Numerical examples are performed to illustrate the effectiveness and the accuracy of the method.

Originality/value

The proposed method is different from the traditional LDG method, which discretes an equation in spatial direction and couples an ordinary differential equation (ODE) solver, such as Runger‐Kutta method. This fully discrete scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. Numerical examples prove that the authors' method is very effective. The present paper is the authors' first step towards an effective approach based on the discontinuous Galerkin method for the solution of fractional‐order problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 14 October 2020

Zhijian Duan and Gongnan Xie

The discontinuous Galerkin finite element method (DGFEM) is very suited for realizing high order resolution approximations on unstructured grids for calculating the hyperbolic…

Abstract

Purpose

The discontinuous Galerkin finite element method (DGFEM) is very suited for realizing high order resolution approximations on unstructured grids for calculating the hyperbolic conservation law. However, it requires a significant amount of computing resources. Therefore, this paper aims to investigate how to solve the Euler equations in parallel systems and improve the parallel performance.

Design/methodology/approach

Discontinuous Galerkin discretization is used for the compressible inviscid Euler equations. The multi-level domain decomposition strategy was used to deal with the computational grids and ensure the calculation load balancing. The total variation diminishing (TVD) Runge–Kutta (RK) scheme coupled with the multigrid strategy was employed to further improve parallel efficiency. Moreover, the Newton Block Gauss–Seidel (GS) method was adopted to accelerate convergence and improve the iteration efficiency.

Findings

Numerical experiments were implemented for the compressible inviscid flow problems around NACA0012 airfoil, over M6 wing and DLR-F6 configuration. The parallel acceleration is near to a linear convergence. The results indicate that the present parallel algorithm can reduce computational time significantly and allocate memory reasonably, which has high parallel efficiency and speedup, and it is well-suited to large-scale scientific computational problems on multiple instruction stream multiple data stream model.

Originality/value

The parallel DGFEM coupled with TVD RK and the Newton Block GS methods was presented for hyperbolic conservation law on unstructured meshes.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 9 September 2019

Yang Xia and Pan Guo

Numerical instability such as spurious oscillation is an important problem in the simulation of heat wave propagation. The purpose of this study is to propose a time discontinuous

Abstract

Purpose

Numerical instability such as spurious oscillation is an important problem in the simulation of heat wave propagation. The purpose of this study is to propose a time discontinuous Galerkin isogeometric analysis method to reduce numerical instability of heat wave propagation in the medium subjected to heat sources, particularly heat impulse.

Design/methodology/approach

The essential vectors of temperature and the temporal gradients are assumed to be discontinuous and interpolated individually in the discretized time domain. The isogeometric analysis method is applied to use its property of smooth description of the geometry and to eliminate the mesh-dependency. An artificial damping scheme with proportional stiffness matrix is brought into the final discretized form to reduce the numerical spurious oscillations.

Findings

The numerical spurious oscillations in the simulation of heat wave propagation are effectively eliminated. The smooth description of geometry with spline functions solves the mesh-dependency problem and improves the numerical precision.

Originality/value

The time discontinuous Galerkin method is applied within the isogeometric analysis framework. The proposed method is effective in the simulation of the wave propagation problems subjecting to impulse load with numerical stability and accuracy.

Details

Engineering Computations, vol. 36 no. 7
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 15 June 2018

Ali Karakus, Tim Warburton, Mehmet Haluk Aksel and Cuneyt Sert

This study aims to focus on the development of a high-order discontinuous Galerkin method for the solution of unsteady, incompressible, multiphase flows with level set interface…

Abstract

Purpose

This study aims to focus on the development of a high-order discontinuous Galerkin method for the solution of unsteady, incompressible, multiphase flows with level set interface formulation.

Design/methodology/approach

Nodal discontinuous Galerkin discretization is used for incompressible Navier–Stokes, level set advection and reinitialization equations on adaptive unstructured elements. Implicit systems arising from the semi-explicit time discretization of the flow equations are solved with a p-multigrid preconditioned conjugate gradient method, which minimizes the memory requirements and increases overall run-time performance. Computations are localized mostly near the interface location to reduce computational cost without sacrificing the accuracy.

Findings

The proposed method allows to capture interface topology accurately in simulating wide range of flow regimes with high density/viscosity ratios and offers good mass conservation even in relatively coarse grids, while keeping the simplicity of the level set interface modeling. Efficiency, local high-order accuracy and mass conservation of the method are confirmed through distinct numerical test cases of sloshing, dam break and Rayleigh–Taylor instability.

Originality/value

A fully discontinuous Galerkin, high-order, adaptive method on unstructured grids is introduced where flow and interface equations are solved in discontinuous space.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 28 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 March 1993

J. PETERA, V. NASSEHI and J.F.T. PITTMAN

A number of finite element formulations involving discontinuous weighting functions have been tested against analytic solutions for a steady scalar convection—diffusion problem at…

Abstract

A number of finite element formulations involving discontinuous weighting functions have been tested against analytic solutions for a steady scalar convection—diffusion problem at intermediate Peclet number, with a ‘hard’ downstream boundary condition. The emphasis is on extending these methods to isoparametric bilinear and biquadratic elements. In order to do this a procedure is given for the exact calculation of shape function Laplacians. Having confirmed the success of the Brooks—Hughes streamline upwind Petrov—Galerkin (SUPG) method for isoparametric bilinear elements, formulations for biquadratic elements are examined. Galerkin least squares offers little advantage over SUPG in the test problem. The generalized Galerkin method of Donea et al. gave excellent results, but because of concern over the possibility of cross‐streamline artificial diffusion in some cases, a strictly streamline formulation incorporating the optimal parameters of Donea et al. is proposed. This gave excellent results on a sufficiently refined mesh.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 3 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 4 January 2016

Mehdi Jamei and H Ghafouri

The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the…

Abstract

Purpose

The purpose of this paper is to present a novel sequential implicit discontinuous Galerkin (DG) method for two-phase incompressible flow in porous media. It is based on the wetting phase pressure-saturation formulation with Robin boundary condition (Klieber and Riviere, 2006) using H(div) velocity projection.

Design/methodology/approach

The local mass conservation and continuity of normal component of velocity across elements interfaces are enforced by a simple H(div) velocity projection in lowest order Raviart-Thomas (RT0) space. As further improvements, the authors use the weighted averages and the scaled penalties in spatial DG discretization. Moreover, the Chavent-Jaffre slope limiter, as a consistent non-oscillatory limiter, is used for saturation values to avoid the spurious oscillations.

Findings

The proposed model is verified by a pseudo 1D Buckley-Leverett problem in homogeneous media. Two homogeneous and heterogeneous quarter five-spot benchmark problems and a random permeable medium are used to show the accuracy of the method at capturing the sharp front and illustrate the impact of proposed improvements.

Research limitations/implications

The work illustrates incompressible two-phase flow behavior and the capillary pressure heterogeneity between different geological layers is assumed to be negligible.

Practical implications

The proposed model can efficiently be used for modeling of two-phase flow in secondary recovery of petroleum reservoirs and tracing the immiscible contamination in porous media.

Originality/value

The authors present an efficient sequential DG method for immiscible incompressible two-phase flow in porous media with improved performance for detection of sharp frontal interfaces and discontinuities.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 December 2005

N. Canouet, L. Fezoui and S. Piperno

The use of the prominent FDTD method for the time domain solution of electromagnetic wave propagation past devices with small geometrical details can require very fine grids and…

Abstract

Purpose

The use of the prominent FDTD method for the time domain solution of electromagnetic wave propagation past devices with small geometrical details can require very fine grids and can lead to very important computational time and storage. The purpose is to develop a numerical method able to handle possibly non‐conforming locally refined grids, based on portions of Cartesian grids in order to use existing pre‐ and post‐processing tools.

Design/methodology/approach

A Discontinuous Galerkin method is built based on bricks and its stability, accuracy and efficiency are proved.

Findings

It is found to be possible to conserve exactly the electromagnetic energy and weakly preserves the divergence of the fields (on conforming grids). For non‐conforming grids, the local sets of basis functions are enriched at subgrid interfaces in order to get rid of possible spurious wave reflections.

Research limitations/implications

Although the dispersion analysis is incomplete, the numerical results are really encouraging it is shown the proposed numerical method makes it possible to handle devices with extremely small details. Further investigations are possible with different, higher‐order discontinuous finite elements.

Originality/value

This paper can be of great value for people wanting to migrate from FDTD methods to more up to date time‐domain methods, while conserving existing pre‐ and post‐processing tools.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 24 no. 4
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 3 May 2013

Liang Li, Stéphane Lanteri and Ronan Perrussel

This work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two‐dimensional…

Abstract

Purpose

This work is concerned with the development and the numerical investigation of a hybridizable discontinuous Galerkin (HDG) method for the simulation of two‐dimensional time‐harmonic electromagnetic wave propagation problems.

Design/methodology/approach

The proposed HDG method for the discretization of the two‐dimensional transverse magnetic Maxwell equations relies on an arbitrary high order nodal interpolation of the electromagnetic field components and is formulated on triangular meshes. In the HDG method, an additional hybrid variable is introduced on the faces of the elements, with which the element‐wise (local) solutions can be defined. A so‐called conservativity condition is imposed on the numerical flux, which can be defined in terms of the hybrid variable, at the interface between neighbouring elements. The linear system of equations for the unknowns associated with the hybrid variable is solved here using a multifrontal sparse LU method. The formulation is given, and the relationship between the considered HDG method and a standard upwind flux‐based DG method is also examined.

Findings

The approximate solutions for both electric and magnetic fields converge with the optimal order of p+1 in L2 norm, when the interpolation order on every element and every interface is p and the sought solution is sufficiently regular. The presented numerical results show the effectiveness of the proposed HDG method, especially when compared with a classical upwind flux‐based DG method.

Originality/value

The work described here is a demonstration of the viability of a HDG formulation for solving the time‐harmonic Maxwell equations through a detailed numerical assessment of accuracy properties and computational performances.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 32 no. 3
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 1 March 1997

Pinhas Z. Bar‐Yoseph and Eduard Moses

Deals with the formulation and application of temporal and spatial spectral element approximations for the solution of convection‐diffusion problems. Proposes a new high‐order…

Abstract

Deals with the formulation and application of temporal and spatial spectral element approximations for the solution of convection‐diffusion problems. Proposes a new high‐order splitting space‐time spectral element method which exploits space‐time discontinuous Galerkin for the first hyperbolic substep and space continuous‐time discontinuous Galerkin for the second parabolic substep. Analyses this method and presents its characteristics in terms of accuracy and stability. Also investigates a subcycling technique, in which several hyperbolic substeps are taken for each parabolic substep; a technique which enables fast, cost‐effective time integration with little loss of accuracy. Demonstrates, by a numerical comparison with other coupled and splitting space‐time spectral element methods, that the proposed method exhibits significant improvements in accuracy, stability and computational efficiency, which suggests that this method is a potential alternative to existing schemes. Describes several areas for future research.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 7 no. 2/3
Type: Research Article
ISSN: 0961-5539

Keywords

1 – 10 of 255