Discontinuous Galerkin method coupled with an integral representation for solving the three-dimensional time-harmonic Maxwell equations

In this paper, we present a mathematical and numerical studies of the three-dimensional time-harmonic Maxwell equations. The problem is solved by a discontinuous Galerkin DG method coupled with an integral representation. This study was completed by some numerical tests to justify the effectiveness of the proposed approach. The numerical simulation was done by an iterative solver implemented in FORTRAN.     

Hybridizable discontinuous Galerkin methods for the time-harmonic Maxwell’s equations

We present two hybridizable discontinuous Galerkin (HDG) methods for the numerical solution of the time-harmonic Maxwell’s equations. The first HDG method explicitly enforces the divergence-free condition and thus necessitates the introduction of a Lagrange multiplier. It produces a linear system for the degrees of freedom of the approximate traces of both the tangential component of the vector field and the Lagrange multiplier. The second HDG method does not explicitly enforce the divergence-free condition and thus results in a linear system for the degrees of freedom of the approximate trace of the tangential component of the vector field only. For both HDG methods, the approximate vector field converges with the optimal order of k + 1 in the L²-norm, when polynomials of degree k are used to represent all the approximate variables. We propose elementwise postprocessing to obtain a new H^curl-conforming approximate vector field which converges with order k + 1 in the H^curl-norm. We present extensive numerical examples to demonstrate and compare the performance of the HDG methods.     

电磁场2.5维时域间断有限元方法

介质沿空间固定方向均匀分布的结构在电磁导波器件中有十分广泛的应用,对这类器件的分析通常被称为2.5D电磁问题。利用器件在固定方向介质分布均匀的特点,将电磁场量沿该方向进行空间傅里叶变换,可以把对三维问题的分析转化为两维问题求解,从而极大地减小计算开销。针对传统基于差分的2.5D电磁场算法在弯曲形状逼近上有阶梯误差的缺陷,本文提出了基于三角形网格的2.5D时域间断有限元方法（DGTD）,并用它模拟了电偶极子与光纤的耦合效率和光子晶体光纤的色散特性。与基于规则网格的2.5D差分方法进行对比。结果表明,文中建立的2.5D DGTD方法对弯曲形状的模拟更加逼真,计算内存占用最大减少10.4%,计算精度最大相差0.011%,计算时间缩短74.9%,计算效率提高。     

New numerical method for solving time-harmonic Maxwell equations with strong singularity

In this paper we introduce a notion of R_ν-generalized solution to time-harmonic Maxwell equations with strong singularity in a 2D nonconvex polygonal domain. We develop a new weighted edge FEM. Results of numerical experiments prove the efficiency of this method.     

Local discontinuous Galerkin method for solving Burgers and coupled Burgers equations

In the current work, we extend the local discontinuous Galerkin method to a more general application system. The Burgers and coupled Burgers equations are solved by the local discontinuous Galerkin method. Numerical experiments are given to verify the efficiency and accuracy of our method. Moreover the numerical results show that the method can approximate sharp fronts accurately with minimal oscillation.     

Locally implicit discontinuous Galerkin method for time domain electromagnetics

In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes.     

Non-conformal domain decomposition method with second-order transmission conditions for time-harmonic electromagnetics

Non-overlapping domain decomposition (DD) methods provide efficient algorithms for solving time-harmonic Maxwell equations. It has been shown that the convergence of DD algorithms can be improved significantly by using high order transmission conditions. In this paper, we extend a newly developed second-order transmission condition (SOTC), which involves two second-order transverse derivatives, to facilitate fast convergence in the non-conformal DD algorithms. However, the non-conformal nature of the DD methods introduces an additional technical difficulty, which results in poor convergence in many real-life applications. To mitigate the difficulty, a corner-edge penalty method is proposed and implemented in conjunction with the SOTC to obtain truly robust solver performance. Numerical results verify the analysis and demonstrate the effectiveness of the proposed methods on a few model problems. Finally, drastically improved convergence, compared to the conventional Robin transmission condition, was observed for an electrically large problem of practical interest.     

Hybridizable discontinuous Galerkin methods for partial differential equations in continuum mechanics

We present hybridizable discontinuous Galerkin methods for solving steady and time-dependent partial differential equations (PDEs) in continuum mechanics. The essential ingredients are a local Galerkin projection of the underlying PDEs at the element level onto spaces of polynomials of degree k to parametrize the numerical solution in terms of the numerical trace; a judicious choice of the numerical flux to provide stability and consistency; and a global jump condition that enforces the continuity of the numerical flux to arrive at a global weak formulation in terms of the numerical trace. The HDG methods are fully implicit, high-order accurate and endowed with several unique features which distinguish themselves from other discontinuous Galerkin methods. First, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Second, they provide, for smooth viscous-dominated problems, approximations of all the variables which converge with the optimal order of k + 1 in the L²-norm. Third, they possess some superconvergence properties that allow us to define inexpensive element-by-element postprocessing procedures to compute a new approximate solution which may converge with higher order than the original solution. And fourth, they allow for a novel and systematic way for imposing boundary conditions for the total stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the methods. In addition, they possess other interesting properties for specific problems. Their approximate solution can be postprocessed to yield an exactly divergence-free and H(div)-conforming velocity field for incompressible flows. They do not exhibit volumetric locking for nearly incompressible solids. We provide extensive numerical results to illustrate their distinct characteristics and compare their performance with that of continuous Galerkin methods.     

A Hermite WENO reconstruction-based discontinuous Galerkin method for the Euler equations on tetrahedral grids

A Hermite WENO reconstruction-based discontinuous Galerkin method RDG(P₁P₂), designed not only to enhance the accuracy of discontinuous Galerkin method but also to ensure linear stability of the RDG method, is presented for solving the compressible Euler equations on tetrahedral grids. In this RDG(P₁P₂) method, a quadratic polynomial solution (P₂) is first reconstructed using a least-squares method from the underlying linear polynomial (P₁) discontinuous Galerkin solution. By taking advantage of handily available and yet invaluable information, namely the derivatives in the DG formulation, the stencils used in the reconstruction involve only von Neumann neighborhood (adjacent face-neighboring cells) and thus are compact and consistent with the underlying DG method. The final quadratic polynomial solution is then obtained using a WENO reconstruction, which is necessary to ensure linear stability of the RDG method. The developed RDG method is used to compute a variety of flow problems on tetrahedral meshes to demonstrate its accuracy, efficiency, robustness, and versatility. The numerical experiments demonstrate that the developed RDG(P₁P₂) method is able to maintain the linear stability, achieve the designed third-order of accuracy: one order accuracy higher than the underlying DG method without significant increase in computing costs and storage requirements.     

Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations

Ideal magnetohydrodynamic (MHD) equations consist of a set of nonlinear hyperbolic conservation laws, with a divergence-free constraint on the magnetic field. Neglecting this constraint in the design of computational methods may lead to numerical instability or nonphysical features in solutions. In our recent work [F. Li, L. Xu, S. Yakovlev, Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, Journal of Computational Physics 230 (2011) 4828–4847], second and third order exactly divergence-free central discontinuous Galerkin methods were proposed for ideal MHD equations. In this paper, we further develop such methods with higher order accuracy. The novelty here is that the well-established H(div)-conforming finite element spaces are used in the constrained transport type framework, and the magnetic induction equations are extensively explored in order to extract sufficient information to uniquely reconstruct an exactly divergence-free magnetic field. The overall algorithm is local, and it can be of arbitrary order of accuracy. Numerical examples are presented to demonstrate the performance of the proposed methods especially when they are fourth order accurate.     

Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

We develop a new hierarchical reconstruction (HR) method  and  for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.     

Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field

In this paper, central discontinuous Galerkin methods are developed for solving ideal magnetohydrodynamic (MHD) equations. The methods are based on the original central discontinuous Galerkin methods designed for hyperbolic conservation laws on overlapping meshes, and use different discretization for magnetic induction equations. The resulting schemes carry many features of standard central discontinuous Galerkin methods such as high order accuracy and being free of exact or approximate Riemann solvers. And more importantly, the numerical magnetic field is exactly divergence-free. Such property, desired in reliable simulations of MHD equations, is achieved by first approximating the normal component of the magnetic field through discretizing induction equations on the mesh skeleton, namely, the element interfaces. And then it is followed by an element-by-element divergence-free reconstruction with the matching accuracy. Numerical examples are presented to demonstrate the high order accuracy and the robustness of the schemes.     

Solving coupled nonlinear Schrodinger equations via a direct discontinuous Galerkin method

In this work,we present the direct discontinuous Galerkin(DDG) method for the one-dimensional coupled nonlinear Schrdinger(CNLS) equation.We prove that the new discontinuous Galerkin method preserves the discrete mass conservations corresponding to the properties of the CNLS system.The ordinary differential equations obtained by the DDG space discretization is solved via a third-order stabilized Runge-Kutta method.Numerical experiments show that the new DDG scheme gives stable and less diffusive results and has excellent long-time numerical behaviors for the CNLS equations.     

Local derivative post-processing for the discontinuous Galerkin method

Obtaining accurate approximations for derivatives is important for many scientific applications in such areas as fluid mechanics and chemistry as well as in visualization applications. In this paper we discuss techniques for computing accurate approximations of high-order derivatives for discontinuous Galerkin solutions to hyperbolic equations related to these areas. In previous work, improvement in the accuracy of the numerical solution using discontinuous Galerkin methods was obtained through post-processing by convolution with a suitably defined kernel. This post-processing technique was able to improve the order of accuracy of the approximation to the solution of time-dependent symmetric linear hyperbolic partial differential equations from order k+1$k+1$ to order 2k+1$2k+1$ over a uniform mesh; this was extended to include one-sided post-processing as well as post-processing over non-uniform meshes. In this paper, we address the issue of improving the accuracy of approximations to derivatives of the solution by using the method introduced by Thomée [19]. It consists in simply taking the α$\alpha$th-derivative of the convolution of the solution with a sufficiently smooth kernel. The order of convergence of the approximation is then independent   of the order of the derivative, |α|$|\alpha |$. We also discuss an efficient way of computing the approximation which does not involve differentiation but the application of simple finite differencing. Our results show that the above-mentioned approximations to the α$\alpha$th-derivative of the exact solution of linear, multidimensional symmetric hyperbolic systems obtained by the discontinuous Galerkin method with polynomials of degree k$k$ converge with order 2k+1$2k+1$ regardless of the order |α|$|\alpha |$ of the derivative.     

Local discontinuous Galerkin methods for the generalized Zakharov system

In this paper we develop a local discontinuous Galerkin (LDG) method for the generalized Zakharov system. Two energy conservations of the LDG scheme are proved for the generalized Zakharov system. Numerical experiments for the Zakharov system are presented to illustrate the accuracy and capability of the methods, including accuracy tests, plane waves, soliton–soliton collisions of the standard and generalized Zakharov system and a two-dimensional problem.     

Continuous and discontinuous Galerkin methods for a scalable three-dimensional nonhydrostatic atmospheric model: Limited-area mode

This paper describes a unified, element based Galerkin (EBG) framework for a three-dimensional, nonhydrostatic model for the atmosphere. In general, EBG methods possess high-order accuracy, geometric flexibility, excellent dispersion properties and good scalability. Our nonhydrostatic model, based on the compressible Euler equations, is appropriate for both limited-area and global atmospheric simulations. Both a continuous Galerkin (CG), or spectral element, and discontinuous Galerkin (DG) model are considered using hexahedral elements. The formulation is suitable for both global and limited-area atmospheric modeling, although we restrict our attention to 3D limited-area phenomena in this study; global atmospheric simulations will be presented in a follow-up paper. Domain decomposition and communication algorithms used by both our CG and DG models are presented. The communication volume and exchange algorithms for CG and DG are compared and contrasted. Numerical verification of the model was performed using two test cases: flow past a 3D mountain and buoyant convection of a bubble in a neutral atmosphere; these tests indicate that both CG and DG can simulate the necessary physics of dry atmospheric dynamics. Scalability of both methods is shown up to 8192 CPU cores, with near ideal scaling for DG up to 32,768 cores.     

Schur-decomposition for 3D matrix equations and its application in solving radiative discrete ordinates equations discretized by Chebyshev collocation spectral method

The Schur-decomposition for three-dimensional matrix equations is developed and used to directly solve the radiative discrete ordinates equations which are discretized by Chebyshev collocation spectral method. Three methods, say, the spectral methods based on 2D and 3D matrix equation solvers individually, and the standard discrete ordinates method, are presented. The numerical results show the good accuracy of spectral method based on direct solvers. The CPU time cost comparisons against the resolutions between these three methods are made using MATLAB and FORTRAN 95 computer languages separately. The results show that the CPU time cost of Chebyshev collocation spectral method with 3D Schur-decomposition solver is the least, and almost only one thirtieth to one fiftieth CPU time is needed when using the spectral method with 3D Schur-decomposition solver compared with the standard discrete ordinates method.