Fourth‐order compact scheme for the one‐dimensional sine‐Gordon equation

Finite difference scheme to the generalized one‐dimensional sine‐Gordon equation is considered in this paper. After approximating the second order derivative in the space variable by the compact finite difference, we transform the sine‐Gordon equation into an initial‐value problem of a second‐order ordinary differential equation. Then Padé approximant is used to approximate the time derivatives. The resulting fully discrete nonlinear finite‐difference equation is solved by a predictor‐corrector scheme. Both Dirichlet and Neumann boundary conditions are considered in our proposed algorithm. Stability analysis and error estimate are given for homogeneous Dirichlet boundary value problems using energy method. Numerical results are given to verify the condition for stability and convergence and to examine the accuracy and efficiency of the proposed algorithm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Solving hyperbolic conservation laws using multiquadric quasi‐interpolation

In this article, we apply the univariate multiquadric (MQ) quasi‐interpolation to solve the hyperbolic conservation laws. At first we construct the MQ quasi‐interpolation corresponding to periodic and inflow‐outflow boundary conditions respectively. Next we obtain the numerical schemes to solve the partial differential equations, by using the derivative of the quasi‐interpolation to approximate the spatial derivative of the differential equation and a low‐order explicit difference to approximate the temporal derivative of the differential equation. Then we verify our scheme for the one‐dimensional Burgers' equation (without viscosity). We can see that the numerical results are very close to the exact solution and the computational accuracy of the scheme is ??(τ), where τ is the temporal step. We can improve the accuracy by using the high‐order quasi‐interpolation. Moreover the methods can be generalized to the other equations. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Generalized average sampling in shift invariant spaces

Average sampling is motivated by realistic needs, e.g., physical limitation of acquisition devices or characteristics of sampling procedure. As an extension of the average sampling, we study generalized average sampling in shift invariant spaces with frame generators, in which averages are taken from suitable channeled version of signals. Illustrative examples are given.     

Singly diagonally implicit runge‐kutta method for time‐dependent reaction‐diffusion equation

In this article, an efficient fourth‐order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge‐Kutta method in time is proposed to solve the time‐dependent one‐dimensional reaction‐diffusion equation. In this scheme, we first approximate the spatial derivative using the second‐order central finite difference then improve it to fourth‐order by applying Padé approximation. A three stage fourth‐order singly diagonally implicit Runge‐Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high‐order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011     

Symmetrical weighted essentially non‐oscillatory‐flux limiter schemes for Hamilton–Jacobi equations

In this paper, we propose a new scheme that combines weighted essentially non‐oscillatory (WENO) procedures together with monotone upwind schemes to approximate the viscosity solution of the Hamilton–Jacobi equations. In one‐dimensional (1D) case, first, we obtain an optimum polynomial on a four‐point stencil. This optimum polynomial is third‐order accurate in regions of smoothness. Next, we modify a second‐order ENO polynomial by choosing an additional point inside the stencil in order to obtain the highest accuracy when combined with the Harten–Osher reconstruction‐evolution method limiter. Finally, the optimum polynomial is considered as a symmetric and convex combination of three polynomials with ideal weights. Following the methodology of the classic WENO procedure, then, we calculate the non‐oscillatory weights with the ideal weights. Numerical experiments in 1D and 2D are performed to compare the capability of the hybrid scheme to WENO schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

Weak solution to a one‐dimensional full compressible non‐Newtonian fluid

This paper is devoted to the existence of global‐in‐time weak solutions to a one‐dimensional full compressible non‐Newtonian fluid. A semi‐discrete finite element scheme is taken to generate approximate solutions, based on an exact projection technique. To enforce convergence of the approximate solutions, the uniform estimate is obtained using an iteration method and energy method, with the help of the weak compactness and convexity. Numerical simulations showing the existence of solutions are presented.     

Phaseless inverse discrete Hilbert transform and determination of signals in shift‐invariant space

In this paper, we first address the uniqueness of phaseless inverse discrete Hilbert transform (phaseless IDHT for short) from the magnitude of discrete Hilbert transform (DHT for short). The measurement vectors of phaseless IDHT do not satisfy the complement property, a traditional requirement for ensuring the uniqueness of phase retrieval. Consequently, the uniqueness problem of phaseless IDHT is essentially different from that of the traditional phase retrieval. For the phaseless IDHT related to compactly supported functions, conditions are given in our first main theorem to ensure the uniqueness. A condition on the step size of DHT is crucial for the insurance. The second main theorem concerns the uniqueness related to noncompactly supported functions. It is not the trivial generalization of the first main theorem. Our third main result is on the determination of signals in shift‐invariant spaces by phaseless IDHT. Note that the measurements used for the determination are the DHT magnitudes. They are the approximations to the Hilbert transform magnitudes. Recall that for the existing methods of phaseless sampling (a special phase‐retrieval problem), to determine a signal depends on the exact measurements but not the approximative ones. Therefore, our determination method is essentially different from the traditional phaseless sampling. Numerical simulations are conducted to examine the efficiency of phaseless IDHT and its application in determining signals in spline Hilbert shift‐invariant space.     

Approximate reconstruction of bandlimited functions for the integrate and fire sampler

In this paper we study the reconstruction of a bandlimited signal from samples generated by the integrate and fire model. This sampler allows us to trade complexity in the reconstruction algorithms for simple hardware implementations, and is specially convenient in situations where the sampling device is limited in terms of power, area and bandwidth. Although perfect reconstruction for this sampler is impossible, we give a general approximate reconstruction procedure and bound the corresponding error. We also show the performance of the proposed algorithm through numerical simulations.     

Sampling theorem and multi-scale spectrum based on non-linear Fourier atoms

This study concerns some new developments of unit analytic signals with non-linear phase. It includes ladder-shaped filter, generalized Sinc function based on non-linear Fourier atoms, generalized sampling theorem for non-bandlimited signals and the notion of multi-scale spectrum for discrete sequences. We first introduce the ladder-shaped filter and show that the impulse response of its corresponding linear time-shift invariant system is the generalized Sinc function as a product of periodic Poisson kernel and Sinc function. Secondly, we establish a Shannon-type sampling theorem based on generalized Sinc function for this type of non-bandlimited signal. We further prove that this type of signal may be holomorphically extended to strips in the complex plane containing the real axis. Finally, we introduce the notion of multi-scale spectrums for discrete sequences and develop the related fast algorithm.     

An Eulerian–Lagrangian Weighted Essentially Nonoscillatory scheme for nonlinear conservation laws

We develop a formally high order Eulerian–Lagrangian Weighted Essentially Nonoscillatory (EL‐WENO) finite volume scheme for nonlinear scalar conservation laws that combines ideas of Lagrangian traceline methods with WENO reconstructions. The particles within a grid element are transported in the manner of a standard Eulerian–Lagrangian (or semi‐Lagrangian) scheme using a fixed velocity v. A flux correction computation accounts for particles that cross the v‐traceline during the time step. If v = 0, the scheme reduces to an almost standard WENO5 scheme. The CFL condition is relaxed when v is chosen to approximate either the characteristic or particle velocity. Excellent numerical results are obtained using relatively long time steps. The v‐traceback points can fall arbitrarily within the computational grid, and linear WENO weights may not exist for the point. A general WENO technique is described to reconstruct to any order the integral of a smooth function using averages defined over a general, nonuniform computational grid. Moreover, to high accuracy, local averages can also be reconstructed. By re‐averaging the function to a uniform reconstruction grid that includes a point of interest, one can apply a standard WENO reconstruction to obtain a high order point value of the function. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 651–680, 2017     

High‐resolution relaxation scheme for the two‐dimensional Riemann problems in gas dynamics

We present a new relaxation method for the numerical approximation of the two‐dimensional Riemann problems in gas dynamics. The novel feature of the technique proposed here is that it does not require either a Riemann solver or a characteristics decomposition. The high resolution of the method is achieved by using a third‐order reconstruction for the space discretization and a third‐order TVD Runge‐Kutta scheme for the time integration. Numerical experiments, using several configurations of Riemann problems in gas dynamics, are included to confirm the high resolution of the new relaxation scheme. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

On harmonic analysis of vector‐valued signals

A vector‐valued signal in N dimensions is a signal whose value at any time instant is an N‐dimensional vector, that is, an element of . The sum of an arbitrary number of such signals of the same frequency is shown to trace an ellipse in N‐dimensional space, that is, to be confined to a plane. The parameters of the ellipse (major and minor axes, represented by N‐dimensional vectors; and phase) are obtained algebraically in terms of the directions of oscillation of the constituent signals, and their phases. It is shown that the major axis of the ellipse can always be determined algebraically. That is, a vector, whose value can be computed algebraically (without decisions or comparisons of magnitude) from parameters of the constituent signals, always represents the major axis of the ellipse. The ramifications of this result for the processing and Fourier analysis of signals with vector values or samples are discussed, with reference to the definition of Fourier transforms, particularly discrete Fourier transforms, such as have been defined in several hypercomplex algebras, including Clifford algebras. The treatment in the paper, however, is entirely based on signals with values in . Although the paper is written in terms of vector signals (which are taken to include images and volumetric images), the analysis clearly also applies to a superposition of simple harmonic motions in N dimensions. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical scheme for solving system of fractional partial differential equations with Volterra‐type integral term through two‐dimensional block‐pulse functions

In the current study, an approximate scheme is established for solving the fractional partial differential equations (FPDEs) with Volterra integral terms via two‐dimensional block‐pulse functions (2D‐BPFs). According to the definitions and properties of 2D‐BPFs, the original problem is transformed into a system of linear algebra equations. By dispersing the unknown variables for these algebraic equations, the numerical solutions can be obtained. Besides, the proof of the convergence of this system is given. Finally, several numerical experiments are presented to test the feasibility and effectiveness of the proposed method.     

Numerical simulation of the nonlinear generalized time‐fractional Klein–Gordon equation using cubic trigonometric B‐spline functions

In this paper, an efficient numerical procedure for the generalized nonlinear time‐fractional Klein–Gordon equation is presented. We make use of the typical finite difference schemes to approximate the Caputo time‐fractional derivative, while the spatial derivatives are discretized by means of the cubic trigonometric B‐splines. Stability and convergence analysis for the numerical scheme are discussed. We apply our scheme to some typical examples and compare the obtained results with the ones found by other numerical methods. The comparison shows that our scheme is quite accurate and can be applied successfully to a variety of problems of applied nature.     

Phaseless Signal Recovery in Infinite Dimensional Spaces Using Structured Modulations

This paper considers the recovery of continuous signals in infinite dimensional spaces from the magnitude of their frequency samples. It proposes a sampling scheme which involves a combination of oversampling and modulations with complex exponentials. Sufficient conditions are given such that almost every signal with compact support can be reconstructed up to a unimodular constant using only its magnitude samples in the frequency domain. Finally it is shown that an average sampling rate of four times the Nyquist rate is enough to reconstruct almost every time-limited signal.     

Detecting inclusions with a generalized impedance condition from electrostatic data via sampling

In this paper, we derive a sampling method to solve the inverse shape problem of recovering an inclusion with a generalized impedance condition from electrostatic Cauchy data. The generalized impedance condition is a second order differential operator applied to the boundary of the inclusion. We assume that the Dirichlet‐to‐Neumann mapping is given from measuring the current on the outer boundary from an imposed voltage. A simple numerical example is given to show the effectiveness of the proposed inversion method for recovering the inclusion. We also consider the inverse impedance problem of determining the impedance parameters for a known material from the Dirichlet‐to‐Neumann mapping assuming the inclusion has been reconstructed where uniqueness for the reconstruction of the coefficients is proven.     

An efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations

In this paper, we propose an efficient spectral‐Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations. Firstly, we carry out a truncation from a three‐dimensional unbounded domain to a bounded spherical domain. By using spherical coordinate transformation and spherical harmonic expansion, we transform the original problem into a series of one‐dimensional eigenvalue problem that can be solved effectively. Secondly, we introduce a weighted Sobolev space to treat the singularity in the effective potential. Using the property of orthogonal polynomials in weighted Sobolev space, the error estimate for the approximate eigenvalues and corresponding eigenfunctions are proved. Error estimates show that our numerical method can achieve spectral accuracy for approximate eigenvalues and eigenfunctions. Finally, we give some numerical examples to demonstrate the efficiency of our algorithms and the correctness of the theoretical results.     

High‐order approximation of 2D convection‐diffusion equation on hexagonal grids

We derive a fourth‐order finite difference scheme for the two‐dimensional convection‐diffusion equation on an hexagonal grid. The difference scheme is defined on a single regular hexagon of size h over a seven‐point stencil. Numerical experiments are conducted to verify the high accuracy of the derived scheme, and to compare it with the standard second‐order central difference scheme. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006     

Stability and convergence of two‐grid Crank‐Nicolson extrapolation scheme for the time‐dependent natural convection equations

In this paper, we consider the Crank‐Nicolson extrapolation scheme for the 2D/3D unsteady natural convection problem. Our numerical scheme includes the implicit Crank‐Nicolson scheme for linear terms and the recursive linear method for nonlinear terms. Standard Galerkin finite element method is used to approximate the spatial discretization. Stability and optimal error estimates are provided for the numerical solutions. Furthermore, a fully discrete two‐grid Crank‐Nicolson extrapolation scheme is developed, the corresponding stability and convergence results are derived for the approximate solutions. Comparison from aspects of the theoretical results and computational efficiency, the two‐grid Crank‐Nicolson extrapolation scheme has the same order as the one grid method for velocity and temperature in H¹‐norm and for pressure in L²‐norm. However, the two‐grid scheme involves much less work than one grid method. Finally, some numerical examples are provided to verify the established theoretical results and illustrate the performances of the developed numerical schemes.     

An enhanced‐physics‐based scheme for the NS‐α turbulence model

We study a new enhanced‐physics‐based numerical scheme for the NS‐alpha turbulence model that conserves both energy and helicity. Although most turbulence models (in the continuous case) conserve only energy, NS‐alpha is one of only a very few that also conserve helicity. This is one reason why it is becoming accepted as the most physically accurate turbulence model. However, no numerical scheme for NS‐alpha, until now, conserved both energy and helicity, and thus the advantage gained in physical accuracy by modeling with NS‐alpha could be lost in a computation. This report presents a finite element numerical scheme, and gives a rigorous analysis of its conservation properties, stability, solution existence, and convergence. A key feature of the analysis is the identification of the discrete energy and energy dissipation norms, and proofs that these norms are equivalent (provided a careful choice of filtering radius) in the discrete space to the usual energy and energy dissipation norms. Numerical experiments are given to demonstrate the effectiveness of the scheme over usual (helicity‐ignoring) schemes. A generalization of this scheme to a family of high‐order NS‐alpha‐deconvolution models, which combine the attractive physical properties of NS‐alpha with the high accuracy gained by combining α‐filtering with van Cittert approximate deconvolution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010