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.     

Finite Element $θ$-Schemes for the Acoustic Wave Equation

In this paper, we investigate the stability and convergence of a family of implicit finite difference schemes in time and Galerkin finite element methods in space for the numerical solution of the acoustic wave equation. The schemes cover the classical explicit second-order leapfrog scheme and the fourth-order accurate scheme in time obtained by the modified equation method. We derive general stability conditions for the family of implicit schemes covering some well-known CFL conditions. Optimal error estimates are obtained. For sufficiently smooth solutions, we demonstrate that the maximal error in the $L^2$-norm error over a finite time interval converges optimally as $\mathcal{O}(h^{p+1}+∆t^s)$, where $p$ denotes the polynomial degree, $s$=2 or 4, $h$ the mesh size, and $∆t$ the time step.     

A second order self-consistent IMEX method for radiation hydrodynamics

We present a second order self-consistent implicit/explicit (methods that use the combination of implicit and explicit discretizations are often referred to as IMEX (implicit/explicit) methods ,  and ) time integration technique for solving radiation hydrodynamics problems. The operators of the radiation hydrodynamics are splitted as such that the hydrodynamics equations are solved explicitly making use of the capability of well-understood explicit schemes. On the other hand, the radiation diffusion part is solved implicitly. The idea of the self-consistent IMEX method is to hybridize the implicit and explicit time discretizations in a nonlinearly consistent way to achieve second order time convergent calculations. In our self-consistent IMEX method, we solve the hydrodynamics equations inside the implicit block as part of the nonlinear function evaluation making use of the Jacobian-free Newton Krylov (JFNK) method ,  and . This is done to avoid order reductions in time convergence due to the operator splitting. We present results from several test calculations in order to validate the numerical order of our scheme. For each test, we have established second order time convergence.     

A discontinuous Galerkin method for inviscid low Mach number flows

In this work we extend the high-order discontinuous Galerkin (DG) finite element method to inviscid low Mach number flows. The method here presented is designed to improve the accuracy and efficiency of the solution at low Mach numbers using both explicit and implicit schemes for the temporal discretization of the compressible Euler equations. The algorithm is based on a classical preconditioning technique that in general entails modifying both the instationary term of the governing equations and the dissipative term of the numerical flux function (full preconditioning approach). In the paper we show that full preconditioning is beneficial for explicit time integration while the implicit scheme turns out to be efficient and accurate using just the modified numerical flux function. Thus the implicit scheme could also be used for time accurate computations. The performance of the method is demonstrated by solving an inviscid flow past a NACA0012 airfoil at different low Mach numbers using various degrees of polynomial approximations. Computations with and without preconditioning are performed on different grid topologies to analyze the influence of the spatial discretization on the accuracy of the DG solutions at low Mach numbers.     

光伏电池组件隐式、显式单二极管模型准确性对比研究

光伏电池组件非线性输出特性的物理建模及其优化参数的准确提取是光伏发电系统设计计算、性能评估及优化控制的重要前提.相对于传统的隐式单二极管模型,该文在光伏电池显式单二极管模型的基础上利用Lambert W函数推导了光伏组件的显式单二极管模型,提出一种基于重启边界约束Nelder-Mead单纯形算法的参数提取方法rbcNM,并利用两种典型光伏电池组件的实测数据对隐式、显式单二极管模型的准确性进行了对比测试和验证.结果表明:rbcNM算法可以快速准确的提取隐式、显式单二极管模型的优化参数,计算结果与实测数据具有很好的一致性,相对于已有文献在准确度上取得了大幅度的提升;显式单二极管模型的准确性显著高于隐式单二极管模型,对光伏电池组件的电流-电压和功率-电压特性曲线具有更高的拟合精度.     

Upwind monotonic interpolation methods for the solution of the time dependent radiative transfer equation

We describe the use of upwind monotonic interpolation methods for the solution of the time-dependent radiative transfer equation in both optically thin and thick media. These methods, originally developed to solve Eulerian advection problems in hydrodynamics, have the ability to propagate sharp features in the flow with very little numerical diffusion. We consider the implementation of both explicit and implicit versions of the method. The explicit version is able to keep radiation fronts resolved to only a few zones wide when higher order interpolation methods are used. Although traditional implementations of the implicit version suffer from large numerical diffusion, we describe an implicit method which considerably reduces this diffusion.     

On the scalar modal analysis of optical waveguides using approximate methods

We have discussed the approximate methods which are used for obtaining scalar guided modes of optical waveguides. The methods include the perturbation method, the variational method including the Rayleigh-Ritz method, and the Galerkin and the collocation method. The main purpose of this paper is to discuss the inter-relationships and equivalences of these methods, and to bring out the fact that these relationships have, in fact, not been recognized in the guided wave optics literature, although in the numerical electromagnetics and applied mathematics literature some of these relationships are well known. We have also pointed out specific examples where, due to this lack of recognition of relationships, there are repetitions in the literature. In particular, we have noted that the Rayleigh-Ritz method and the Galerkin method have been used using the same set of basis functions for the same kind of waveguides without recognizing the existing literature. We have also reported for the first time an explicit relationship between the Galerkin method and the collocation method. This relationship also points out in which cases one method is more accurate and/or numerically efficient than the other. Another interesting relationship explored is that between the perturbation method and the variational method.     

Three-Step Predictor-Corrector of Exponential Fitting Method for Nonlinear Schroedinger Equations

We develop the three-step explicit and implicit schemes of exponential fitting methods. We use the three- step explicit exponential fitting scheme to predict an approximation, then use the three-step implicit exponential fitting scheme to correct this prediction. This combination is called the three-step predictor-corrector of exponential fitting method. The three-step predictor-corrector of exponential fitting method is applied to numerically compute the coupled nonlinear Schroedinger equation and the nonlinear Schroedinger equation with varying coefficients. The numerical results show that the scheme is highly accurate.     

The Locally Conservative Galerkin (LCG) Method — a Discontinuous Methodology Applied to a Continuous Framework

This paper presents a comprehensive overview of the element-wise locally conservative Galerkin (LCG) method. The LCG method was developed to find a method that had the advantages of the discontinuous Galerkin methods, without the large computational and memory requirements. The initial application of the method is discussed, to the simple scalar transient convection-diffusion equation, along with its extension to the Navier-Stokes equations utilising the Characteristic Based Split (CBS) scheme. The element-by-element solution approach removes the standard finite element assembly necessity, with an face flux providing continuity between these elemental subdomains. This face flux provides explicit local conservation and can be determined via a simple small post-processing calculation. The LCG method obtains a unique solution from the elemental contributions through the use of simple averaging. It is shown within this paper that the LCG method provides equivalent solutions to the continuous (global) Galerkin method for both steady state and transient solutions. Several numerical examples are provided to demonstrate the abilities of the LCG method.     

Weighted-Residual Methods for the Solution of Two-Particle Lippmann–Schwinger Equation Without Partial-Wave Decomposition

Recently there has been a growing interest in computational methods for quantum scattering equations that avoid the traditional decomposition of wave functions and scattering amplitudes into partial waves. The aim of the present work is to show that the weighted-residual approach in combination with local basis functions give rise to convenient computational schemes for the solution of the multi-variable integral equations without the partial wave expansion. The weighted-residual approach provides a unifying framework for various variational and degenerate-kernel methods for integral equations of scattering theory. Using a direct-product basis of localized quadratic interpolation polynomials, Galerkin, collocation and Schwinger variational realizations of the weighted-residual approach have been implemented for a model potential. It is demonstrated that, for a given expansion basis, Schwinger variational method exhibits better convergence with basis size than Galerkin and collocation methods. A novel hybrid-collocation method is implemented with promising results as well.     

Particle simulations of space weather

We review the application of particle simulation techniques to the full kinetic study of space weather events. We focus especially on the methods designed to overcome the difficulties created by the tremendous range of time and space scales present in the physical systems. We review the aspects of the derivation of the particle in cell (PIC) method relevant to the discussion. We consider first the explicit formulation highlighting its severe limitations due to the presence of stability constraints. Next we introduce implicit methods designed to remove such constraints. We describe both fully implicit methods based on the use of non-linear iteration solvers and semi-implicit methods based on the linearization of the coupling and on simpler linear solvers. We focus the discussion on the implicit moment method but remark its differences from the direct implicit method. The application of adaptive methods within PIC is discussed. Finally practical considerations about the implementation of the implicit PIC method on massively parallel computers to conduct studies of space weather events are given.     

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier–Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge–Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection–diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier–Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.     

A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems

We present a fully second order implicit/explicit time integration technique for solving hydrodynamics coupled with nonlinear heat conduction problems. The idea is to hybridize an implicit and an explicit discretization in such a way to achieve second order time convergent calculations. In this scope, the hydrodynamics equations are discretized explicitly making use of the capability of well-understood explicit schemes. On the other hand, the nonlinear heat conduction is solved implicitly. Such methods are often referred to as IMEX methods [2], [1], [3]. The Jacobian-Free Newton Krylov (JFNK) method (e.g. [10], [9]) is applied to the problem in such a way as to render a nonlinearly iterated IMEX method. We solve three test problems in order to validate the numerical order of the scheme. For each test, we established second order time convergence. We support these numerical results with a modified equation analysis (MEA) [21], [20]. The set of equations studied here constitute a base model for radiation hydrodynamics.     

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier–Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge–Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection–diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier–Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.     

Energy stable numerical methods for hyperbolic partial differential equations using overlapping domain decomposition

Overlapping domain decomposition methods, otherwise known as overset grid or chimera methods, are useful for simplifying the discretization of partial differential equations in or around complex geometries. Though in wide use, such methods are prone to numerical instability unless numerical diffusion or some other form of regularization is used, especially for higher-order methods. To address this shortcoming, high-order, provably energy stable, overlapping domain decomposition methods are derived for hyperbolic initial boundary value problems. The overlap is treated by splitting the domain into pieces and using generalized summation-by-parts derivative operators and polynomial interpolation. New implicit and explicit operators are derived that do not require regularization for stability in the linear limit. Applications to linear and nonlinear problems in one and two dimensions are presented, where it is found the explicit operators are preferred to the implicit ones.     

Multisymplectic implicit and explicit methods for Klein–Gordon–Schrödinger equations

We propose multisymplectic implicit and explicit Fourier pseudospectral methods for the Klein-Gordon-Schrödinger equations. We prove that the implicit method satisfies the charge conservation law exactly. Both methods provide accurate solutions in long-time computations and simulate the soliton collision well. Numerical results show the abilities of the two methods in preserving charge, energy, and momentum conservation laws.     

三机械薄膜腔光力系统相互作用的研究

本文主要研究了利用传输矩阵理论和共振透射条件详细地推导光腔中均匀放置三个机械薄膜构成的腔光力系统中系统本征模式随机械运动的色散关系.计算结果发现系统的光学本征模式由一组四个的本征能级构成,且不同的能级随不同的机械运动模式的变化曲线各不相同,进而导致不同光学模式与不同机械运动模式之间的耦合也不相同.此外,利用微扰理论求解了当机械运动振幅远小于腔模波长、机械振子处于平衡位置附近时,各种光学模式与不同机械振动模式间相互作用耦合强度的解析表达式.研究结果能够为理论和实验上研究多模腔光力系统提供一定的参考.     

An analysis of time discretization in the finite element solution of hyperbolic problems

The problem of the time discretization of hyperbolic equations when finite elements are used to represent the spatial dependence is critically examined. A modified equation analysis reveals that the classical, second-order accurate, time-stepping algorithms, i.e., the Lax-Wendroff, leap-frog, and Crank-Nicolson methods, properly combine with piecewise linear finite elements in advection problems only for small values of the time step. On the contrary, as the Courant number increases, the numerical phase error does not decrease uniformly at all wavelengths so that the optimal stability limit and the unit CFL property are not achieved. These fundamental numerical properties can, however, be recovered, while still remaining in the standard Galerkin finite element setting, by increasing the order of accuracy of the time discretization. This is accomplished by exploiting the Taylor series expansion in the time increment up to the third order before performing the Galerkin spatial discretization using piecewise linear interpolations. As a result, it appears that the proper finite element equivalents of second-order finite difference schemes are implicit methods of incremental type having third- and fourth-order global accuracy on uniform meshes (Taylor-Galerkin methods). Numerical results for several linear examples are presented to illustrate the properties of the Taylor-Galerkin schemes in one- and two-dimensional calculations.     

Implicit DG Method for Time Domain Maxwell's Equations Involving Metamaterials

An implicit discontinuous Galerkin method is introduced to solve the time-domain Maxwell's equations in metamaterials. The Maxwell's equations in metamaterials are represented by integral-differential equations. Our scheme is based on discontinuous Galerkin method in spatial domain and Crank-Nicolson method in temporal domain. The fully discrete numerical scheme is proved to be unconditionally stable. When polynomial of degree at most $p$ is used for spatial approximation, our scheme is verified to converge at a rate of $\mathcal{O}(τ^2+h^{p+1/2})$. Numerical results in both 2D and 3D are provided to validate our theoretical prediction.