Approximations of steady periodic water waves in flows with constant vorticity

We provide high-order approximations to periodic travelling wave profiles and to the velocity field and the pressure beneath the waves, in flows with constant vorticity over a flat bed.     

On surface radiation conditions for high-frequency wave scattering

A new approximation of the logarithmic derivative of the Hankel function is derived and applied to high-frequency wave scattering. We re-derive the on surface radiation condition (OSRC) approximations that are well suited for a Dirichlet boundary in acoustics. These correspond to the Engquist–Majda absorbing boundary conditions. Inverse OSRC approximations are derived and they are used for Neumann boundary conditions. We obtain an implicit OSRC condition, where we need to solve a tridiagonal system. The OSRC approximations are well suited for moderate wave numbers. The approximation of the logarithmic derivative is also used for deriving a generalized physical optics approximation, both for Dirichlet and Neumann boundary conditions. We have obtained similar approximations in electromagnetics, for a perfect electric conductor. Numerical computations are done for different objects in 2D and 3D and for different wave numbers. The improvement over the standard physical optics is verified.     

Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

Sorin Micu ^{This paper studies the numerical approximation of the boundarycontrol for the wave equation in a square domain. It is knownthat the discrete and semi-discrete models obtained by discretizingthe wave equation with the usual finite-difference or finite-elementmethods do not provide convergent sequences of approximationsto the boundary control of the continuous wave equation as themesh size goes to zero. Here, we introduce and analyse a newsemi-discrete model based on the space discretization of thewave equation using a mixed finite-element method with two differentbasis functions for the position and velocity. The main theoreticalresult is a uniform observability inequality which allows usto construct a sequence of approximations converging to theminimal L2-norm control of the continuous wave equation. Wealso introduce a fully discrete system, obtained from our semi-discretescheme, for which we conjecture that it provides a convergentsequence of discrete approximations as both h and t, the timediscretization parameter, go to zero. We illustrate this factwith several numerical experiments.}     

On the preservation of invariants in the simulation of solitary waves in some nonlinear dispersive equations

In this paper we generalize some results in the literature concerning the structure of numerical approximations to solitary wave solutions of some nonlinear, dispersive equations is studied. We prove that those time discretizations with the property of preserving, exactly or approximately up to certain order, some invariants of the problems, have a better propagation of the error and provide a more suitable simulation of the solitary waves. The generalization involves the treatment of nonlocal operators and two different kinds of equations.     

Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation

This study is devoted to recovering two initial values for a time‐fractional diffusion‐wave equation from boundary Cauchy data. We provide the uniqueness result for recovering two initial values simultaneously by the method of Laplace transformation and analytic continuation. And then we use a nonstationary iterative Tikhonov regularization method to solve the inverse problem and propose a finite dimensional approximation algorithm to find good approximations to the initial values. Numerical examples in one‐ and two‐dimensional cases are provided to show the effectiveness of the proposed method.     

A well-balanced approximate Riemann solver for compressible flows in variable cross-section ducts

A well-balanced approximate Riemann solver is introduced in this paper in order to compute approximations of one-dimensional Euler equations in variable cross-section ducts. The interface Riemann solver is grounded on the VFRoe-ncv scheme, and it enforces the preservation of Riemann invariants of the steady wave. The main properties of the scheme are detailed. We provide numerical results to assess the validity of the scheme, even when the cross-section is discontinuous. A first series is devoted to analytical test cases, and the last results correspond to the simulation of a bubble collapse.     

Delta shock wave formation in the case of triangular hyperbolic system of conservation laws

We describe δ-shock wave generation from continuous initial data in the case of triangular conservation law system arising from “generalized pressureless gas dynamics model.” We use smooth approximations in the weak sense that are more general than small viscosity approximations.     

Coulomb Scattering of a Slow Quantum Particle in a Space of Arbitrary Dimension

We assume that a charged quantum particle moves in a space of dimension d = 2, 3,... and is scattered by a fixed Coulomb center. We derive and study expansions of the wave function and all radial functions of such a particle in integer powers of the wave number and in Bessel functions of a real order. We prove that finite sums of such expansions are asymptotic approximations of the wave functions in the low-energy limit.     

Diffraction of a plane wave by a circular disk

In this paper we solve the problem of diffraction of a normally incident plane wave by a circular disk. We treat both the hard and soft disk. In each case we obtain the solution as a series which converges when the product of the wave number and the radius of the disk is large. Our construction leads directly to asymptotic approximations to the solution for large wave number.     

Resonances in bounded media: Nonlinear and boundary effects

We study the response of nonlinear wave systems in bounded domains at or near resonance. There are typically two qualitatively distinct types of response which may be observed relating to whether or not higher harmonics are themselves resonant. We introduce a variety of nonlinear model problems at or near resonance and study the subsequent response. We explain how the features of this problem such as the form of nonlinearity, boundary conditions, and the nature of spectrum play a fundamental role in the qualitative nature of the response. Numerical simulations are carried out to provide further explanation and comparison with analytic approximations. The results of this study provide a better understanding of the impact and interplay between nonlinear and boundary effects and thus in turn will contribute to providing new insights into various physically motivated problems in acoustics and other settings.     

The Trotter-Kato theorem and approximation of PDEs

We present formulations of the Trotter-Kato theorem for approximation of linear C-semigroups which provide very useful framework when convergence of numerical approximations to solutions of PDEs are studied. Applicability of our results is demonstrated using a first order hyperbolic equation, a wave equation and Stokes' equation as illustrative examples.

     

Nonlinear dispersive wave problems

In this paper we consider a certain class of nonlinear dispersive wave problems having solutions in the form of slowly varying wavetrains. We develop a procedure generating successively formal asymptotic approximations of these wavetrains of increasing asymptotic accuracy. In order to obtain formal asymptotic approximations we apply the two variable construction technique as developed in [3] for a class of perturbed oscillations described by nonlinear ordinary differential equations containing a small nonnegative perturbation parameter ?.     

A priori estimates for two multiscale finite element methods using multiple global fields to wave equations

We consider a scalar wave equation with nonseparable spatial scales. If the solution of the wave equation smoothly depends on some global fields, then we can utilize the global fields to construct multiscale finite element basis functions. We present two finite element approaches using the global fields: partition of unity method and mixed multiscale finite element method. We derive a priori error estimates for the two approaches and theoretically investigate the relation between the smoothness of the global fields and convergence rates of the approximations for the wave equation. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Two-grid discretization schemes for nonlinear Schrödinger equations

We study efficient two-grid discretization schemes with two-loop continuation algorithms for computing wave functions of two-coupled nonlinear Schrödinger equations defined on the unit square and the unit disk. Both linear and quadratic approximations of the operator equations are exploited to derive the schemes. The centered difference approximations, the six-node triangular elements and the Adini elements are used to discretize the PDEs defined on the unit square. The proposed schemes also can compute stationary solutions of parameter-dependent reaction–diffusion systems. Our numerical results show that it is unnecessary to perform quadratic approximations.     

Modulated High-Frequency Waves

We construct asymptotic approximations to solutions of nonlinear hyperbolic conservation laws, when the initial data is small-amplitude high-frequency waves with modulated wave number. We show that the nonlinear multiwave interaction terms approach zero in the asymptotic limit, so that the wave components satisfy decoupled Burgers equations, provided a certain nonresonance condition holds. This extends previous results on more strictly nonresonant or everywhere resonant waves, to permit modulated high frequencies to pass through resonant and nonresonant values. We show how these results apply to high-frequency wave propagation in a nonhomogeneous medium or on a nonuniform base state. We illustrate our conclusions with numerical examples and discuss a phenomenon of localized resonance.     

Global weak solutions to the Novikov equation by viscous approximation

We study global weak solutions to the Novikov equation by vanishing viscosity method. We prove that global weak solutions can be obtained as weak limits of viscous approximations for a class of initial data. The proof relies on a space–time higher integrability estimate and the method of renormalization. In addition, we analyze the interaction of peakon and antipeakon and prove that wave breaking leads to energy concentration. By different continuations beyond the wave breaking, we obtain conservative solutions and dissipative solutions respectively.     

二维空间中半线性摄动波动方程初值问题解的渐近理论

研究二维空间中具初值问题的半线性波动方程解的渐近理论,在二次连续的古典空间中得到了形式近似解的渐近合理性在长时间范围内成立,这一结果描述了渐近解的长时间存在性.作为所得到的渐近理论的应用,对二维空间中的一个特殊波动方程作出了分析.     

Sobolev estimates for constructive uniform-grid FFT interpolatory approximations of spherical functions

The fast Fourier transform (FFT) based matrix-free ansatz interpolatory approximations of periodic functions are fundamental for efficient realization in several applications. In this work we design, analyze, and implement similar constructive interpolatory approximations of spherical functions, using samples of the unknown functions at the poles and at the uniform spherical-polar grid locations \(\left (\frac {j\pi }{N}, \frac {k \pi }{N}\right )\), for j=1,…,N?1, k=0,…,2N?1. The spherical matrix-free interpolation operator range space consists of a selective subspace of two dimensional trigonometric polynomials which are rich enough to contain all spherical polynomials of degree less than N. Using the \({\mathcal {O}}(N^{2})\) data, the spherical interpolatory approximation is efficiently constructed by applying the FFT techniques (in both azimuthal and latitudinal variables) with only \({\mathcal {O}}(N^{2} \log N)\) complexity. We describe the construction details using the FFT operators and provide complete convergence analysis of the interpolatory approximation in the Sobolev space framework that are well suited for quantification of various computer models. We prove that the rate of spectrally accurate convergence of the interpolatory approximations in Sobolev norms (of order zero and one) are similar (up to a log term) to that of the best approximation in the finite dimensional ansatz space. Efficient interpolatory quadratures on the sphere are important for several applications including radiation transport and wave propagation computer models. We use our matrix-free interpolatory approximations to construct robust FFT-based quadrature rules for a wide class of non-, mildly-, and strongly-oscillatory integrands on the sphere. We provide numerical experiments to demonstrate fast evaluation of the algorithm and various theoretical results presented in the article.     

Numerical analysis of finite dimensional approximations of Kohn–Sham models

In this paper, we study finite dimensional approximations of Kohn–Sham models, which are widely used in electronic structure calculations. We prove the convergence of the finite dimensional approximations and derive the a priori error estimates for ground state energies and solutions. We also provide numerical simulations for several molecular systems that support our theory.     

Using multiple objectives to approximate normative models

This paper discusses the rationale for the use of additive models involving multiple objectives as approximations to normative analyses. Experience has shown us that organizations often evaluate important decisions with multiple objective models rather than reducing all aspects of the problem to a single criterion, dollars, as many normative economic models prescribe. We justify this practice on two grounds: managers often prefer to think about a problem in terms of several dimensions and a multiple objective model may provide an excellent approximation to the more complex normative model. We argue that a useful analysis based on a multiple objective model will fulfill both conditions—it will provide insights for the decision maker as well as a good approximation to the normative model. We report several real-world examples of managers using multiple objective models to approximate such normative models as the risk-adjusted net present value and the value of information models. The agreement between the approximate models and the normative models is shown to be quite good. Next, we cite a portion of the behavioral decision theory literature which establishes that linear models of multiple attributes provide quite robust approximations to individual decision-making processes. We then present more general theoretical and empirical results which support our contention that linear multiple attribute models can provide good approximations to more complex models.