New Predictor-Corrector Methods Based on Exponential Time Differencing for Systems of Nonlinear Differential Equations

We present the new predictor-corrector methods for systems of nonlinear differential equations, based on the method of exponential time differencing. We compare the present schemes with the explicit multistep exponential time differencing and Adams-Bashforth-Moulton method. The numerical results show that the schemes are more accurate and more efficient than Adams predictor-corrector method. The exponential time differencing method has been developed and perfected by the present studies.     

任意阶隐式指数时程差分多步法及其在非线性系统中的应用

对非线性系统提出了任意阶隐式指数时程差分多步法，实现了任意阶次指数时程差分预测 校正算法.发展完善了指数时程差分法.将新算法应用于非线性系统，取得了较好的效果.数值结果表明隐式指数时程差分多步法很好地修正了显式指数时程差分多步法，隐式指数时程差分多步法是一种高精度、高效率的方法. 关键词： 非线性系统 任意阶隐式指数时程差分多步法 混沌     

Exact Solutions to Nonlinear Schroedinger Equation and Higher-Order Nonlinear Schroedinger Equation

We study solutions of the nonlinear Schroedinger equation （NLSE） and higher-order nonlinear Sehroedinger equation （HONLSE） with variable coefficients. By considering all the higher-order effect of HONLSE as a new dependent variable, the NLSE and HONLSE can be changed into one equation. Using the generalized Lie group reduction method （CLGRM）, the abundant solutions of NLSE and HONLSE are obtained.     

Explicit and implicit finite difference schemes for fractional Cattaneo equation

In this paper, the numerical solution of fractional (non-integer)-order Cattaneo equation for describing anomalous diffusion has been investigated. Two finite difference schemes namely an explicit predictor–corrector and totally implicit schemes have been developed. In developing each scheme, a separate formulation approach for the governing equations has been considered. The explicit predictor–corrector scheme is the fractional generalization of well-known MacCormack scheme and has been called Generalized MacCormack scheme. This scheme solves two coupled low-order equations and simultaneously computes the flux term with the main variable. Fully implicit scheme however solves a single high-order undecomposed equation. For Generalized MacCormack scheme, stability analysis has been studied through Fourier method. Through a numerical test, the experimental order of convergency of both schemes has been found. Then, the domain of applicability and some numerical properties of each scheme have been discussed.     

A dispersion-relation-preserving algorithm for a nonlinear shallow-water wave equation

The paper presents an iterative algorithm for studying a nonlinear shallow-water wave equation. The equation is written as an evolution equation, involving only first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step iterative method that first solves the evolution equation by the implicit midpoint rule and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The midpoint rule is a symplectic time integrator for Hamiltonian systems, which may be a preferable method to solve the spatially discretized evolution equation. To give an assessment of the dispersion-preserving scheme, we provide a detailed analysis of the dispersive and dissipative errors of this algorithm. Via a variety of examples, we illustrate the efficiency and accuracy of the proposed scheme by examining the errors in different norms and providing the rates of convergence of the method. In addition, we provide several examples to demonstrate that the conserved quantities of the equation are well preserved by the implicit midpoint time integrator. Finally, we compare the accuracy, elapsed computing time, and spatial and temporal rates of convergence among the proposed method, a complete integrable particle method, and the local discontinuous Galerkin method.     

空间分数阶导数“反常”扩散方程数值算法的比较

分别采用显式差分格式、隐式差分格式以及Crank-Nicholson差分格式数值求解空间分数阶导数,并从局部截断误差、稳定性、计算量三个方面进行比较分析;通过数值算例验证分析结果.     

Numerical solution of the one-dimensional Burgers’ equation: Implicit and fully implicit exponential finite difference methods

This paper describes two new techniques which give improved exponential finite difference solutions of Burgers’ equation. These techniques are called implicit exponential finite difference method and fully implicit exponential finite difference method for solving Burgers’ equation. As the Burgers’ equation is nonlinear, the scheme leads to a system of nonlinear equations. At each time-step, Newton’s method is used to solve this nonlinear system. The results are compared with exact values and it is clearly shown that results obtained using both the methods are precise and reliable.     

反应扩散模型在图灵斑图中的应用及数值模拟

反应扩散方程模型常被用于描述生物学中斑图的形成.从反应扩散模型出发,理论推导得到GiererMeinhardt模型的斑图形成机理,解释了非线性常微分方程系统的稳定常数平衡态在加入扩散项后会发生失稳并产生图灵斑图的过程.通过计算该模型,得到图灵斑图产生的参数条件.数值方法中采用一类有效的高精度数值格式,即在空间离散条件下采用Chebyshev谱配置方法,在时间离散条件下采用紧致隐积分因子方法.该方法结合了谱方法和紧致隐积分因子方法的优点,具有精度高、稳定性好、存储量小等优点.数值模拟表明,在其他条件一定的情况下,系统控制参数κ取不同值对于斑图的产生具有重要的影响,数值结果验证了理论结果.     

超导低电阻接头衰减法测试过程中的拟合偏差控制

超导技术在许多高新技术领域有着广泛而深远的应用前景,其中超导导体接头技术是超导发展过程中必不可少的关键环节之一,快速而又准确地评估低电阻接头显得尤为重要.本文在考虑超导体的E-J特性后,针对目前低电阻接头采用的衰减法测量,计算得出感应电流完整的衰减曲线方程-幂函数方程.对比目前广泛采用的指数方程拟合与采用幂函数方程拟合的偏差,指出采用指数方程代替幂函数方程拟合的必要条件,为接头电阻衰减法测试,尤其是低电阻接头情况下的测量提供参考.     

Painleve Integrability of Nonlinear SchrSdinger Equations with both Space- and Time-Dependent Coefficients

We investigate the Painleve integrabiiity of nonautonomous nonlinear Schr6dinger （NLS） equations with both space-and time-dependent dispersion, nonlinearity, and external potentials. The Painleve analysis is carried out without using the Kruskal＇s simplification, which results in more generalized form of inhomogeneous equations. The obtained equations are shown to be reducible to the standard NLS equation by using a point transformation. We also construct the corresponding Lax pair and carry out its Kundu-type reduction to the standard Lax pair. Special cases of equations from choosing limited form of coefficients coincide with the equations from the previous Painleve analyses and/or become unknown new equations.     

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.     

Compact implicit integration factor methods for some complex-valued nonlinear equations

The compact implicit integration factor (cIIF) method is an efficient time discretization scheme for stiff nonlinear diffusion equations in two and three spatial dimensions. In the current work, we apply the cIIF method to some complex-valued nonlinear evolutionary equations such as the nonlinear Schro¨dinger (NLS) equation and the complex Ginzburg–Landau (GL) equation. Detailed algorithm formulation and practical implementation of cIIF method are performed. The numerical results indicate that this method is very accurate and efficient.     

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.     

Propagation of local spatial solitons in power-law nonlinear PT-symmetric potentials based on finite difference

We consider the (2+1)-dimensional nonlinear Schrödinger equation with power-law nonlinearity under the parity-time-symmetry potential by using the Crank–Nicolson alternating direction implicit difference scheme, which can also be used to solve general boundary problems under the premise of ensuring accuracy. We use linear Fourier analysis to verify the unconditional stability of the scheme. To demonstrate the effectiveness of the scheme, we compare the numerical results with the exact soliton solutions. Moreover, by using the scheme, we test the stability of the solitons under the small environmental disturbances.     

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.     

Implicit Versus Explicit Renormalization of the NN Force: An S-Wave Toy Model

We use an S-wave toy model for the two-nucleon system to show that the implicit renormalization of a contact theory matches the explicit renormalization through a flow equation which integrates out the high momentum components. By fitting the low-momentum interaction with a new contact theory, we show that the running of the contact strengths in both original and fitted contact theories match over a wide cutoff range.     

Self-Similar Solutions of Variable-Coefficient Cubic-Quintic Nonlinear Schroedinger Equation with an External Potential

An improved homogeneous balance principle and self-similar solutions to the cubic-quintic nonlinear Schroedinger and impose constraints on the functions describing dispersion, self-similar waves are presented.     

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.     

粒子输运离散纵标方程基于界面修正的并行计算方法

为了改造粒子输运方程求解的隐式格式,研究设计适应大型并行计算机的并行计算方法,介绍一类求解粒子输运方程离散纵标方程组的基于界面修正的源迭代并行计算方法.应用空间区域分解,在子区域内界面处首先采用迎风显式差分格式进行预估,构造子区域的入射边界条件,然后,在各个子区域内部进行源迭代求解隐式离散纵标方程组.在源迭代过程中,在内界面入射边界处采用隐式格式进行界面修正.数值算例表明该并行计算方法在精度、并行度、简单性诸方面均具有良好的性质.     

The Gibbons–Tsarev equation: symmetries,invariant solutions,and applications

In this paper we present the full classification of symmetry-invariant solutions for the Gibbons–Tsarev equation. Then we use these solutions to construct explicit expressions for two-component reductions of Benney’s moments equations, to get solutions of Pavlov’s equation, and to find integrable reductions of the Ferapontov–Huard–Zhang system, which describes implicit two-phase solutions of the dKP equation.