An exponential transformation based splitting method for fast computations of highly oscillatory solutions

Splitting, or decomposition, methods have been widely used for achieving higher computational efficiency in solving wave equations. A major concern has remained, however, if the wave number involved is exceptionally large. In the case, merits of a conventional splitting method may diminish due to the fact that tiny discretization steps need to be employed to compensate high oscillations. This paper studies an alternative way for solving highly oscillatory paraxial wave problems via a modified splitting strategy. In the process, an exponential transformation is first introduced to convert the underlying differential equation to coupled nonlinear equations. Then the resulted oscillation-free system is treated by a Local-One-Dimensional (LOD) scheme for desired accuracy, efficiency and computability. The splitting method acquired is asymptotically stable and easy to use. Computational experiments are given to illustrate our numerical procedures.     

Stroboscopic averaging of highly oscillatory nonlinear wave equations

In this paper, we are concerned with stroboscopic averaging for highly oscillatory evolution equations posed in a Banach space. Using Taylor expansion, we construct a non‐oscillatory high‐order system whose solution remains exponentially close to the exact one over a long time. We then apply this result to the nonlinear wave equation in one dimension. We present the stroboscopic averaging method, which is a numerical method introduced by Chartier, Murua and Sanz‐Serna, and apply it to our problem. Finally, we conclude by presenting the qualitative and quantitative efficiency of this numerical method for some nonlinear wave problem. Copyright © 2014 John Wiley & Sons, Ltd.     

固体中短波传播的单位分解有限元法

提出了固体中短波传播数值模拟的单位分解有限元法．有限元空间由形成单位分解的标准等参有限元形函数乘以定义为局部子空间基函数的特殊形函数构成．特殊形函数使试空间中包含了关于波动方程的已有知识,因而在单个单元内能近似地再现高度振荡性质．数值例题显示了所提出单位分解有限元在计算精度和效率上的良好性能．     

Numerical methods for oscillatory solutions to hyperbolic problems

Difference approximations of hyperbolic partial differential equations with highly oscillatory coefficients and initial values are studied. Analysis of strong and weak convergence is carried out in the practically interesting case when the discretization step sizes are essentially independent of the oscillatory wave lengths. © 1993 John Wiley & Sons, Inc.     

Long-Time Oscillatory Energy Conservation of Total Energy-Preserving Methods for Highly Oscillatory Hamiltonian Systems

For an integrator when applied to a highly oscillatory system,the near conservation of the oscillatory energy over long times is an important aspect.In this paper,we study the long-time near conservation of oscillatory energy for the adapted average vector field(AAVF)method when applied to highly oscillatory Hamiltonian systems.This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly.This paper is devoted to analysing another important property of AAVF method,i.e.,the near conservation of its oscillatory energy in a long term.The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion.A similar result of the method in the multi-frequency case is also presented in this paper.     

Note on the homotopy perturbation method for multivariate vector-value oscillatory integrals

This paper considers a homotopy perturbation method for approximating multivariate vector-value highly oscillatory integrals. The asymptotic formulae of the integrals and the asymptotic order of the asymptotic method are presented. Numerical examples show the efficiency of the approximation method.     

Efficient finite-difference multi-scheme approach to the simulation of seismic waves in anisotropic media

This paper presents an original multi-scheme approach to the numerical simulation of seismic wave propagation in models with anisotropic formations. To simulate wave propagation in the anisotropic parts of the model, the Lebedev scheme is used. This scheme is rather universal, but highly expensive in terms of computational efficiency. In the main part of the model, a highly efficient standard staggered grid scheme is proposed. The two schemes are coupled to ensure convergence of the reflection/propagation coefficients with a prescribed order. The algorithm combines the universality of the Lebedev scheme and the efficiency of the standard staggered grid scheme.     

Efficient integration for a class of highly oscillatory integrals

This paper presents some quadrature methods for a class of highly oscillatory integrals whose integrands may have singularities at the two endpoints of the interval. One is a Filon-type method based on the asymptotic expansion. The other is a Clenshaw-Curtis-Filon-type method which is based on a special Hermite interpolation polynomial and can be evaluated efficiently in O(N log N) operations, where N + 1 is the number of Clenshaw-Curtis points in the interval of integration. In addition, we derive the corresponding error bound in inverse powers of the frequency ω for the Clenshaw-Curtis-Filon-type method for the class of highly oscillatory integrals. The efficiency and the validity of these methods are testified by both the numerical experiments and the theoretical results.     

Sobolev smoothing of SVD-based Fourier continuations

A method for calculating Sobolev smoothed Fourier continuations is presented. The method is based on the recently introduced singular value decomposition based Fourier continuation approach. This approach allows for highly accurate Fourier series approximations of non-periodic functions. These super-algebraically convergent approximations can be highly oscillatory in an extended region, contaminating the Fourier coefficients. It is shown that through solving a subsequent least squares problem, a Fourier continuation can be produced which has been dramatically smoothed in that the Fourier coefficients exhibit a prescribed rate of decay as the wave number increases. While the smoothing procedure has no significant negative effect on the accuracy of the Fourier series approximation, in some situations the smoothed continuations can actually yield increased accuracy in the approximation of the function and its derivatives.     

Surface integral methods for high-frequency electromagnetic scattering

Surface integral equation based methods are advantageous for simulation of electromagnetic waves scattered by three dimensional obstacles, because they efficiently reduce the dimension of the problem and are robust for high-frequency problems. However, the cost of setting up the associated discretized dense linear systems is prohibitive due to evaluation of highly oscillatory magnetic and electric dipole surface integral operators using standard cubatures. The computational complexity of evaluating such integrals depends on the incident wave frequency, and the size and shape of the obstacles. In this work we discuss a surface integral reformulation of the scattering problem that involves evaluation of surface integrals with a highly oscillatory physical density, and discuss methods for efficient evaluation of such integrals for a class of smooth three dimensional scatterers whose diameter is a large multiple of the incident wavelength. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

MsFEM `a la Crouzeix-Raviart for Highly Oscillatory Elliptic Problems

We introduce and analyze a multiscale finite element type method (MsFEM) in the vein of the classical Crouzeix-Raviart finite element method that is specifically adapted for highly oscillatory elliptic problems. We illustrate numerically the efficiency of the approach and compare it with several variants of MsFEM.     

Qualitative analysis and solutions of bounded traveling wave for B-BBM equation

In this paper, we apply the theory of planar dynamical systems to carry out qualitative analysis for the dynamical system corresponding to B-BBM equation, and obtain global phase portraits under various parameter conditions. Then, the relations between the behaviors of bounded traveling wave solutions and the dissipation coeffiicient μ are investigated. We find that a bounded traveling wave solution appears as a kink profile solitary wave solution when μ is more than the critical value obtained in this paper, while a bounded traveling wave solution appears as a damped oscillatory solution when μ is less than it. Furthermore, we explain the solitary wave solutions obtained in previous literature, and point out their positions in global phase portraits. In the meantime, approximate damped oscillatory solutions are given by means of undetermined coefficients method. Finally, based on integral equations that reflect the relations between the approximate damped oscillatory solutions and the implicit exact damped oscillatory solutions, error estimates for the approximate solutions are presented.     

Approximate damped oscillatory solutions for compound KdV-Burgers equation and their error estimates

In this paper,we focus on studying approximate solutions of damped oscillatory solutions of the compound KdV-Burgers equation and their error estimates.We employ the theory of planar dynamical systems to study traveling wave solutions of the compound KdV-Burgers equation.We obtain some global phase portraits under different parameter conditions as well as the existence of bounded traveling wave solutions.Furthermore,we investigate the relations between the behavior of bounded traveling wave solutions and the dissipation coefficient r of the equation.We obtain two critical values of r,and find that a bounded traveling wave appears as a kink profile solitary wave if |r| is greater than or equal to some critical value,while it appears as a damped oscillatory wave if |r| is less than some critical value.By means of analysis and the undetermined coefficients method,we find that the compound KdV-Burgers equation only has three kinds of bell profile solitary wave solutions without dissipation.Based on the above discussions and according to the evolution relations of orbits in the global phase portraits,we obtain all approximate damped oscillatory solutions by using the undetermined coefficients method.Finally,using the homogenization principle,we establish the integral equations reflecting the relations between exact solutions and approximate solutions of damped oscillatory solutions.Moreover,we also give the error estimates for these approximate solutions.     

Convergence analysis of non-conforming Trigonometric Finite Wave Elements

Trigonometric Finite Wave Elements (TFWE) are finite elements for solving problems in computational optics. The solution of those problems consist of highly oscillatory waves. TFWE are designed for obtaining optimal approximation properties for such kinds of waves with a changing wave number k. In this article, we study the convergence properties of 2-dimensional non-conforming TFWE by applying Strang’s Lemma.     

一类高振荡Hilbert变换的计算

Hilbert变换在信号处理与医学图像处理中都有着广泛的应用,但是对于一般的含有高振荡因子且在积分区阍中包含奇异值的‰变换往往处理起来较为困难,本文提出了一种基于等距节点插值的高效计算方法。     

Weakly Nonlocal Solitary Waves in a Singularly Perturbed Nonlinear Schrödinger Equation

We consider the nonlinear Schrödinger equation perturbed by the addition of a third-derivative term whose coefficient constitutes a small parameter. It is known from the work of Wai et al. [1] that this singular perturbation causes the solitary wave solution of the nonlinear Schrödinger equation to become nonlocal by the radiation of small-amplitude oscillatory waves. The calculation of the amplitude of these oscillatory waves requires the techniques of exponential asymptotics. This problem is re-examined here and the amplitude of the oscillatory waves calculated using the method of Borel summation. The results of Wai et al. [1] are modified and extended.     

Qualitative analysis and approximate damped oscillatory solutions for a kind of nonlinear dispersive-dissipative equation

This paper makes qualitative analysis to the bounded traveling wave solutions for a kind of nonlinear dispersive-dissipative equation, and considers its solving problem. The relation is investigated between behavior of its solution and the dissipation coefficient. Further, all approximate damped oscillatory solutions when dissipation coefficient is small are presented by utilizing the method of undetermined coefficients according to the theory of rotated vector field in planar dynamical systems. Finally, error estimate is given by establishing the integral equation which reflects the relation between approximate and exact damped oscillatory solutions applying the idea of homogenization principle.     

Three-dimensional gravity waves in a channel of variable depth

We consider existence of three-dimensional gravity waves traveling along a channel of variable depth. It is well known that the long-wave small-amplitude expansion for such waves results in the stationary Korteweg–de Vries equation, coefficients of which depend on the transverse topography of the channel. This equation has a single-humped solitary wave localized in the direction of the wave propagation. We show, however, that there exists an infinite set of resonant Fourier modes that travel at the same speed as the solitary wave does. This fact suggests that the solitary wave confined in a channel of variable depth is always surrounded by small-amplitude oscillatory disturbances in the far-field profile.     

Transversal line defects induced by an electric field in a period-2 oscillatory medium

The dynamical behavior of spiral waves in a period-2 oscillatory medium is investigated under the influence of an external applied alternating current field. Open and closed transversal line defects which wiggle along the direction parallel to the wave fronts, are generated in the spiral-wave patterns when the stimulus frequency of the electric field is equal to one, three or five times of the local oscillatory frequency in the period-2 state. Their generations are directly related with the change in the spatial wavelength induced by the electric field. These wigglings proliferate along the transverse direction parallel to the wave fronts as the stimulus strength increases, and become denser when the stimulus frequency increases by multiples of the period-2 oscillatory frequency.