A predictor–corrector scheme for the improved Boussinesq equation

An implicit finite-difference method based on rational approximants of second order to the matrix-exponential term in a three-time level recurrence relation has been proposed for the numerical solution of the improved Boussinesq equation already known from the bibliography. The method, which is analyzed for local truncation error and stability, leads to the solution of a nonlinear system. To overcome this difficulty a predictor–corrector (P–C) scheme in which the predictor is also a second order implicit one is proposed. The efficiency of the proposed method is tested to various wave packets and the results arising from the experiments are compared with the relevant ones known in the bibliography.     

Quartic B-spline Galerkin approach to the numerical solution of the KdVB equation

The nonlinear Korteweg-de Vries–Burgers’ equation is solved numerically by method of Galerkin using quartic B-splines as both shape and weight functions over the finite intervals. Five test problems are studied to demonstrate the accuracy and efficiency of the proposed method. A comparison of numerical results of both algorithm and some published articles is done in computational section. The numerical results are found in good agreement with exact solutions.     

The piezoelectric continuum

Existence, uniqueness and periodic solutions for the elliptic–hyperbolic system of piezoelectric elastic bodies are studied. In the stationary case the proof of existence and uniqueness is based on the Lax–Milgram lemma defining a suitable bilinear form to take into account the lack of global ellipticity. The linear initial-boundary value problem is dealt with using the theory of semigroup. Certain similarities with the wave equation are also discussed. A related problem that is non-linear for the presence of a damping term is also treated, and a result of existence and uniqueness proved using the Galerkin method. Mathematics Subject Classification (2000) 75F15, 74G50     

A Fourier spectral-discontinuous Galerkin method for time-dependent 3-D Schrödinger–Poisson equations with discontinuous potentials

In this paper, we propose a high order Fourier spectral-discontinuous Galerkin method for time-dependent Schrödinger–Poisson equations in 3-D spaces. The Fourier spectral Galerkin method is used for the two periodic transverse directions and a high order discontinuous Galerkin method for the longitudinal propagation direction. Such a combination results in a diagonal form for the differential operators along the transverse directions and a flexible method to handle the discontinuous potentials present in quantum heterojunction and supperlattice structures. As the derivative matrices are required for various time integration schemes such as the exponential time differencing and Crank Nicholson methods, explicit derivative matrices of the discontinuous Galerkin method of various orders are derived. Numerical results, using the proposed method with various time integration schemes, are provided to validate the method.     

半直线上BBM方程初边值问题的修正有理谱耦合方法

本文考虑使用修正的有理谱方法处理半直线上的BBM方程初边值问题.对非线性项使用Chebyshev有理插值显式处理,而线性项使用修正Legendre有理谱方法隐式处理.这种处理既可以节约运算又可以保持良好的稳定性.数值例子表明了算法的有效性     

A space–time discontinuous Galerkin method for linear convection-dominated Sobolev equations

This article presents a space–time discontinuous Galerkin (DG) finite element method for linear convection-dominated Sobolev equations. The finite element method has basis functions that are continuous in space and discontinuous in time, and variable spatial meshes and time steps are allowed. In the discrete intervals of time, using properties of the Radau quadrature rule, eliminates the restriction to space–time meshes of convectional space–time Galerkin methods. The existence and uniqueness of the approximate solution are proved. An optimal priori error estimate in L^∞(H¹) is derived. Numerical experiments are presented to confirm theoretical results.     

A higher-order moment method of the lattice Boltzmann model for the conservation law equation

In this paper, we proposed a higher-order moment method in the lattice Boltzmann model for the conservation law equation. In contrast to the lattice Bhatnagar–Gross–Krook (BGK) model, the higher-order moment method has a wide flexibility to select equilibrium distribution function. This method is based on so-called a series of partial differential equations obtained by using multi-scale technique and Chapman–Enskog expansion. According to Hirt’s heuristic stability theory, the stability of the scheme can be controlled by modulating some special moments to design the third-order dispersion term and the fourth-order dissipation term. As results, the conservation law equation is recovered with higher-order truncation error. The numerical examples show the higher-order moment method can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the conservation law equation.     

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

Summary A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.     

A piecewise constant collocation method using cosine mesh grading for Symm's equation

Summary Here we study the piecewise constant collocation method using mesh grading to solve Symm's integral equation on [–1, 1]. We give a mesh grading for which this method achieves the optimal order of convergence even though the piecewise constant Galerkin method with the same mesh grading does not. Some numerical results are given.     

A discontinuous Galerkin method for the Wigner-Fokker-Planck equation with a non-polynomial approximation space

A discontinuous Galerkin approach to the Wigner-Fokker-Planck equation, a model for quantum devices including environmental effects, is proposed. Evaluation of a pseudo-differential term is the main challenge. The approach can be applied to a variety of potential functions and uses general approximation spaces. Simulations using the method are in agreement with established analytic results and produce reasonable solutions for several potentials. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

三维非线性抛物型积分-微分方程的A.D.I.Galerkin方法及其分析

研究三维非线性抛物型积分-微分方程的A.D.I.Galerkin方法.通过交替方向,化三维为一维,简化计算;通过Galerkin法,保持高精度.成功处理了Volterra项的影响;对所提Galerkin及A.D.I.Galerkin格式给出稳定性和收敛性分析,得到最佳H1和L2模估计.     

定常不可压Navier—Stokes方程迎风格式

本文利用齐次定解条件对定常不可压Navier—Stokes方程的非线性项进行处理,给出了相应的一种迎风Galerkin有限元算法;针对这种迎风Galerkin有限元算法,在迎风参数满足一定条件下,利用其三项式具有的一些很好性质,更简单地证明了该问题解的存在唯一性。     

Dynamic Hemivariational Inequalities and Their Applications

Dynamic hemivariational inequalities are studied in the present paper. Starting from their solution in the distributional sense, we give certain existence and approximation results by using the Faedo–Galerkin method and certain compactness arguments. Applications from mechanics (viscoelasticity) illustrate the theory.     

An isogeometric discontinuous Galerkin method for Euler equations

An isogeometric discontinuous Galerkin method for Euler equations is proposed. It integrates the idea of isogeometric analysis with the discontinuous Galerkin framework by constructing each element through the knots insertion and degree elevation techniques in non‐uniform rational B‐splines. This leads to the solution inherently shares the same function space as the non‐uniform rational B‐splines representation, and results in that the curved boundaries as well as the interfaces between neighboring elements are naturally and exactly resolved. Additionally, the computational cost is reduced in contrast to that of structured grid generation. Numerical tests demonstrate that the presented method can be high order of accuracy and flexible in handling curved geometry. Copyright © 2016 John Wiley & Sons, Ltd.     

Discrete Wavelet Petrov–Galerkin Methods

In this paper, we develop a discrete wavelet Petrov–Galerkin method for integral equations of the second kind with weakly singular kernels suitable for solving boundary integral equations. A compression strategy for the design of a fast algorithm is suggested. Estimates for the rate of convergence and computational complexity of the method are provided.     

3Development of methods of solving nonlinear boundary problems of stability of viscoelastic thin-walled elements

A new numerical method of solving nonlinear boundary problems based on collocational-type approximations is developed with examples of one- and two-dimensional stability problems for a viscoelastic shell. The method allows us to simplify significantly the arithmetization procedure of differential expressions with subsequent efficient computer implementation of the algorithm. A high degree of calculation accuracy is provided. The estimation of the efficiency of the new numerical scheme is obtained by comparing the results with data where the system was modeled using the method of finite differences and the Galerkin method.Rostov State University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 1, pp. 51–58, January–February, 1999.     

Methods of exploiting natural internal stresses to strengthen composites

A method of strengthening nonuniformly reinforced composites is proposed. A rational scheme for coordinating the external stress field, the resistance field, and the internal stress field is examined in relation to the case when the internal stresses are caused by shrinkage of the resin.Ural Kirov Polytechnic Institute, Sverdlovsk. Translated from Mekhanika Polimerov, No. 5, pp. 870–875, September–October, 1970.     

Uniformly Convergent Multigrid Methods for Convection–Diffusion Problems without Any Constraint on Coarse Grids

We construct a class of multigrid methods for convection–diffusion problems. The proposed algorithms use first order stable monotone schemes to precondition the second order standard Galerkin finite element discretization. To speed up the solution process of the lower order schemes, cross-wind-block reordering of the unknowns is applied. A V-cycle iteration, based on these algorithms, is then used as a preconditioner in GMRES. The numerical examples show that this method is convergent without imposing any constraint on the coarsest grid and the convergence of the preconditioned method is uniform.     

Comparison of generalized differential quadrature and Galerkin methods for the analysis of micro-electro-mechanical coupled systems

This paper presents a comprehensive comparison study between the generalized differential quadrature (GDQ) and the well-known global Galerkin method for analysis of pull-in behavior of nonlinear micro-electro-mechanical coupled systems. The nonlinear governing integro-differential equation for double clamped MEMS devices which was derived using variational principle by the authors [Sadeghian H, Rezazadeh G, Osterberg PM. Application of the generalized differential quadrature method to the study of pull-in phenomena of MEMS switches. J Microelectromech Syst 2007;16(6):1334–40] is discretized by applying Galerkin and GDQ methods. The divergence instability or pull-in phenomenon is analyzed. Obtained results are compared with the results of the pervious works. The Galerkin method is implemented with effect of number of used shape functions. Different types of trail functions on calculated pull-in voltage are examined.Furthermore, compare to one term and two terms truncation Galerkin method, it is observed that the GDQ with small number of grid points (non-uniform) performs accurate results for nonlinear micro-electro-mechanical coupled behavior which requires a large number of grid points at high-order approximation.     

Weak Galerkin finite element methods for Parabolic equations

A newly developed weak Galerkin method is proposed to solve parabolic equations. This method allows the usage of totally discontinuous functions in approximation space and preserves the energy conservation law. Both continuous and discontinuous time weak Galerkin finite element schemes are developed and analyzed. Optimal‐order error estimates in both H¹ and L² norms are established. Numerical tests are performed and reported. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013