多辛Dirac方程的高阶整体保能量格式

基于四阶平均向量场方法和拟谱方法构造了Dirac方程的高阶整体保能量格式，利用构造的高阶整体保能量格式数值模拟方程孤立波的演化行为.数值模拟结果表明构造的高阶整体保能量格式可以很好地模拟Dirac方程孤立波的演化行为，并且可以精确地保持方程的整体能量守恒特性.     

High order finite difference method for time-space fractional differential equations with Caputo and Riemann-Liouville derivatives

We consider high order finite difference methods for two-dimensional fractional differential equations with temporal Caputo and spatial Riemann-Liouville derivatives in this paper. We propose a scheme and show that it converges with second order in time and fourth order in space. The accuracy of our proposed method can be improved by Richardson extrapolation. Approximate solution is obtained by the generalized minimal residual (GMRES) method. A preconditioner is proposed to improve the efficiency for the implementation of the GMRES method.     

Convergence of a non-stiff boundary integral method for interfacial flows with surface tension

Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics. This makes the spatial discretization even more difficult and, at the same time, imposes a severe time step constraint for stable explicit time integration methods.

A proof of the convergence of a reformulated boundary integral method for two-density fluid interfaces with surface tension is presented. The method is based on a scheme introduced by Hou, Lowengrub and Shelley [ J. Comp. Phys. 114 (1994), pp. 312-338] to remove the high order stability constraint or stiffness. Some numerical filtering is applied carefully at certain places in the discretization to guarantee stability. The key of the proof is to identify the most singular terms of the method and to show, through energy estimates, that these terms balance one another.

The analysis is at a time continuous-space discrete level but a fully discrete case for a simple Hele-Shaw interface is also studied. The time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regularity of the solution.

The robustness of the method is illustrated with several numerical examples. A numerical simulation of an unstably stratified two-density interfacial flow shows the roll-up of the interface; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method remains stable even in the full nonlinear regime of motion. Another application of the method shows the process of drop formation in a falling single fluid.

     

High Order Energy-Preserving Method of the "Good" Boussinesq Equation

The fourth order average vector field (AVF) method is applied to solve the "Good" Boussinesq equation. The semi-discrete system of the "good" Boussinesq equation obtained by the pseudo-spectral method in spatial variable, which is a classical finite dimensional Hamiltonian system, is discretized by the fourth order average vector field method. Thus, a new high order energy conservation scheme of the "good" Boussinesq equation is obtained. Numerical experiments confirm that the new high order scheme can preserve the discrete energy of the "good" Boussinesq equation exactly and simulate evolution of different solitary waves well.     

Linear B-spline finite element method for the improved Boussinesq equation

In this paper, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the nonlinear partial differential equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem to which many accurate numerical methods are readily applicable. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.     

On per-iteration complexity of high order Chebyshev methods for sparse functions with banded Hessians

Some algorithms for unconstrained and differentiable optimization problems involve the evaluation of quantities related to high order derivatives. The cost of these evaluations depends widely on the technique used to obtain the derivatives and on some characteristics of the objective function: its size, structure and complexity. Functions with banded Hessian are a special case that we study in this paper. Because of their partial separability, the cost of obtaining their high order derivatives, subtly computed by the technique of automatic differentiation, makes High order Chebyshev methods more interesting for banded systems than for dense functions. These methods have an attractive efficiency as we can improve their convergence order without increasing significantly their algorithmic costs. This paper provides an analysis of the per-iteration complexities of High order Chebyshev methods applied to sparse functions with banded Hessians. The main result can be summarized as: the per-iteration complexity of a High order Chebyshev method is of order of the objective function’s. This theoretical analysis is verified by numerical illustrations.     

任意精度的三点紧致显格式及其在CFD中的应用

通过在泰勒级数展开中运用逐阶迭代的方法,推导出了空间任意精度的三点紧致显格式的表达式,又由Fourier分析法得到了格式的数值弥散和耗散特性．与以往的高精度紧致差分格式不同,提出的格式不用隐式求解代数方程组并且可以达到任意精度．通过方波问题和顶盖方腔流的算例表明,格式在稀疏网格下可以得到很高的精度,不仅能节省计算量,而且易于编程,有很高的计算效率．     

一维高精度离散GDQ方法

GDQ method is a kind of high order accurate numerical methods developed several years ago, which have been successfully used to simulate the solution of smooth engineering problems such as structure mechanics and incompressible fluid dynamics. In this paper, extending the traditional GDQ method, we develop a new kind of discontinuous GDQ methods to solve compressible flow problems of which solutions may be discontinuous. In order to capture the local features of fluid flows, firstly, the computational domain is divided into many small pieces of subdomains. Then, in each small subdomain, the GDQ method is implementedand some kinds of numerical flux limitation conditions will be required to keep the correct flow direction. At the boundary interface between subdomains, we also use some kind of flux conditions according to the flow direction. The numerical method obtained by the above steps has the advantages of high order accuracy and easy to treat boundary conditions. It can simulate perfectly nonlinear waves such as shock, rarefaction wave and contact discontinuity. Finally, the numerical experiments on one dimensional Burgers equation and Euler equations are given.The numerical results verify the validation of the method.     

Wavelets and high order numerical differentiation

Numerical differentiation is a classical ill-posed problem. In this paper, a wavelet regularization method for high order numerical derivatives is given and an order optimal Hölder-type stability estimate is also provided. Some numerical examples show that the method is very effective.     

An efficient meshfree point collocation moving least squares method to solve the interface problems with nonhomogeneous jump conditions

We are going to study a simple and effective method for the numerical solution of the closed interface boundary value problem with both discontinuities in the solution and its derivatives. It uses a strong‐form meshfree method based on the moving least squares (MLS) approximation. In this method, for the solution of elliptic equation, the second‐order derivatives of the shape functions are needed in constructing the global stiffness matrix. It is well‐known that the calculation of full derivatives of the MLS approximation, especially in high dimensions, is quite costly. In the current work, we apply the diffuse derivatives using an efficient technique. In this technique, we calculate the higher‐order derivatives using the approximation of lower‐order derivatives, instead of calculating directly derivatives. This technique can improve the accuracy of meshfree point collocation method for interface problems with nonhomogeneous jump conditions and can efficiently estimate diffuse derivatives of second‐ and higher‐orders using only linear basis functions. To introduce the appropriate discontinuous shape functions in the vicinity of interface, we choose the visibility criterion method that modifies the support of weight function in MLS approximation and leads to an efficient computational procedure for the solution of closed interface problems. The proposed method is applied for elliptic and biharmonic interface problems. For the biharmonic equation, we use a mixed scheme, which replaces this equation by a coupled elliptic system. Also the application of the present method to elasticity equation with discontinuities in the coefficients across a closed interface has been provided. Representative numerical examples demonstrate the accuracy and robustness of the proposed methodology for the closed interface problems. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1031–1053, 2015     

Numerical solution of a system of fourth order boundary value problems using variational iteration method

In this paper, the variational iteration method is used to solve a system of fourth order boundary value problems associated with obstacle, unilateral and contact problems. Numerical solution obtained by the method is of high accuracy. Moreover, the higher-order derivatives of numerical solution can also approximate the higher-order derivatives of exact solution well. Five examples compared with those considered by Siddiqi and Akram [S.S. Siddiqi, G. Akram, Numerical solution of a system of fourth order boundary value problems using cubic non-polynomial spline method, Applied Mathematics and Computation 190 (2007) 652–661] show that the method is more efficient.     

Approximations of choice probabilities in mixed logit models

This paper is concerned with the approximate computation of choice probabilities in mixed logit models. The relevant approximations are based on the Taylor expansion of the classical logit function and on the high order moments of the random coefficients. The approximate choice probabilities and their derivatives are used in conjunction with log likelihood maximization for parameter estimation. The resulting method avoids the assumption of an apriori distribution for the random tastes. Moreover experiments with simulation data show that it compares well with the simulation based methods in terms of computational cost.     

One-step 9-stage Hermite–Birkhoff–Taylor DAE solver of order 10

The HBT(10)9 method for ODEs is expanded into HBT(10)9DAE for solving nonstiff and moderately stiff systems of fully implicit differential algebraic equations (DAEs) of arbitrarily high fixed index. A scheme to generate first-order derivatives at off-step points is combined with Pryce scheme which generates high order derivatives at step points. The stepsize is controlled by a local error estimator. HBT(10)9DAE uses only the first four derivatives of y instead of the first 10 required by Taylor’s series method T10DAE of order 10. Dormand–Prince’s DP(8,7)13M for ODEs is extended to DP(8,7)DAE for DAEs. HBT(10)9DAE wins over DP(8,7)DAE on several test problems on the basis of CPU time as a function of relative error at the end of the interval of integration. An index-5 problem is equally well solved by HBT(10)9DAE and T10DAE. On this problem, the error in the solution by DP(8,7)DAE increases as time increases.     

Derivation of transitional asymptotic expansions from integral representations

A method for deriving transitional asymptotic expansions from integral representations is described and applied to Anger function and modified Hankel function. The method consists in deriving asymptotic expansions of the function considered as well as its first derivativeat the transition point using conventional methods such as Laplace’s method or the method of steepest descents. Since both the functions considered satisfy a second order linear differential equation, it is possible to obtain asymptotic expansions of higher order derivatives of the functions from the first two expansions. Thus asymptotic expressions for all the derivatives at the transition point are known and a Taylor expansion of the function in the neighbourhood of the transition point can be written. The method is also applicable to the generalized exponential integral, Weber’s parabolic cylinder function and Poiseuille function.     

Finite differences and collocation methods for the solution of the two‐dimensional heat equation

In this article, we combine finite difference approximations (for spatial derivatives) and collocation techniques (for the time component) to numerically solve the two‐dimensional heat equation. We employ, respectively, second‐order and fourth‐order schemes for the spatial derivatives, and the discretization method gives rise to a linear system of equations. We show that the matrix of the system is nonsingular. Numerical experiments carried out on serial computers show the unconditional stability of the proposed method and the high accuracy achieved by the fourth‐order scheme. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 54–63, 2001     

Theoretical study of a Bénard-Marangoni problem

In this paper we prove the existence of strong solutions for the stationary Bénard-Marangoni problem in a finite domain flat on the top, bifurcating from the basic heat conductive state. The Bénard-Marangoni problem is a physical phenomenon of thermal convection in which the effects of buoyancy and surface tension are taken into account. This problem is modelled with a system of partial differential equations of the type Navier-Stokes and heat equation. The boundary conditions include crossed boundary conditions involving tangential derivatives of the temperature and normal derivatives of the velocity field. To define tangential derivatives at the boundary, intended in the trace sense, it is necessary order two derivatives in the interior of the domain and thus the boundary term contains as high derivatives as the interior term. We overcome this difficulty by considering the weak formulation, and transforming the boundary integral into an equivalent integral defined in the whole domain. This allows us to reformulate the weak problem with a temperature having only order one weak derivatives. Concerning regularity results, we obtain strong solutions for the stationary Bénard-Marangoni problem.     

Cahn-Hilliard方程的高阶保能量散逸性方法

能量散逸性是物理和力学中某些微分方程一项重要的物理特性.构造精确地保持微分方程能量散逸性的数值格式对模拟具有能量散逸性的微分方程具有重要的意义.本文利用四阶平均向量场方法和傅里叶谱方法构造了Cahn-Hilliard方程高阶保能量散逸性格式.数值结果表明高阶保能量散逸性格式能很好地模拟Cahn-Hilliard方程在不同初始条件下解的行为,并且很好地保持了Cahn-Hilliard方程的能量散逸特性.     

Superconvergence and Asymptotic Expansions for Bilinear Finite Volume Element Approximations

Aiming at the isoparametric bilinear finite volume element scheme, we initially derive an asymptotic expansion and a high accuracy combination formula of the derivatives in the sense of pointwise by employing the energy-embedded method on uniform grids. Furthermore, we prove that the approximate derivatives are convergent of order two. Finally, numerical examples verify the theoretical results.     

An algorithm for the finite difference approximation of derivatives with arbitrary degree and order of accuracy

In this paper, we introduce an algorithm and a computer code for numerical differentiation of discrete functions. The algorithm presented is suitable for calculating derivatives of any degree with any arbitrary order of accuracy over all the known function sampling points. The algorithm introduced avoids the labour of preliminary differencing and is in fact more convenient than using the tabulated finite difference formulas, in particular when the derivatives are required with high approximation accuracy. Moreover, the given Matlab computer code can be implemented to solve boundary-value ordinary and partial differential equations with high numerical accuracy. The numerical technique is based on the undetermined coefficient method in conjunction with Taylor’s expansion. To avoid the difficulty of solving a system of linear equations, an explicit closed form equation for the weighting coefficients is derived in terms of the elementary symmetric functions. This is done by using an explicit closed formula for the Vandermonde matrix inverse. Moreover, the code is designed to give a unified approximation order throughout the given domain. A numerical differentiation example is used to investigate the validity and feasibility of the algorithm and the code. It is found that the method and the code work properly for any degree of derivative and any order of accuracy.     

On a high order numerical method for functions with singularities

By splitting a given singular function into a relatively smooth part and a specially structured singular part, it is shown how the traditional Fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals. Singular functions with various types of singularities of importance in applications are considered. Relations between the discrete and the continuous Fourier series for the singular functions are established. Of particular interest are piecewise smooth functions, for which various important applications are indicated, and for which numerous numerical results are presented.