Narrow Results

In this paper, we propose a new type of finite difference weighted essentially nonoscillatory (WENO) schemes to approximate the viscosity solutions of the Hamilton–Jacobi equations. The new scheme has three properties: (1) the scheme is fifth-order accurate in smooth regions while keep sharp discontinuous transitions with no spurious oscillations near discontinuities; (2) the linear weights can be any positive numbers with the symmetry requirement and that their sum equals one; (3) the scheme can avoid the clipping of extrema. Extensive numerical examples are provided to demonstrate the accuracy and the robustness of the proposed scheme.

In this paper, a high-order Recursive Quadratic (RQ) algorithm is introduced, and the features of this algorithm are thoroughly studied and illustrated. In addition, a robust RQ algorithm is also developed in the presence of general bounded noisy data to enhance model robustness. A numerical example is included to demonstrate the efficiency of RQ algorithms by comparing the results with both the projection algorithm and the conventional recursive least squares algorithm. Also some simulations are carried out to illustrate the effectiveness of the RQ and robust RQ algorithms.

The current work describes the application of high-order numerical techniques to single or multiple overset curvilinear body-fitted grids, demonstrating the feasibility of direct computations of noise radiated by flows around complex non-Cartesian bodies. Flows of both physical and industrial interest can be investigated with this approach. We first rapidly describe the numerical techniques implemented in our curvilinear simulations. The explicit high-order differencing and filtering schemes are presented, as well as their application to the curvilinear Navier–Stokes equations. We then present brief results of various 2-D acoustic simulations. First the flow around a cylinder, and the associated acoustic field, are described. The diameter-based Reynolds number Re_D = 150 is under the critical Reynolds number of the onset of 3-D phenomena in the vortex-shedding. Simulation results can thus be meaningfully compared to experimental measurements. A case of acoustic scattering is then examined. A non-compact monopolar source is placed half way between two differently sized cylinders. A complex diffraction pattern is created, and resulting RMS pressure data are compared to the analytical solution. Finally the noise generated by a low Reynolds number laminar flow around a NACA 0012 airfoil is presented.

The scalar wave equation in a two-dimensional semi-infinite wave guide is considered. The recently proposed Hagstrom–Warburton (H–W) local high-order absorbing boundary conditions (ABCs), which are based on a modification of the Higdon ABCs, are presented in this context. The P-order ABC involves the free parameters 0 < a_j ≤ 1, for j = 0, 1, …, P, which have to be chosen. The choice a_j = 1 for all j is shown to be satisfactory, in general, although not necessarily optimal. The optimal choice of the parameters is discussed via both theoretical analysis and numerical experiments. In addition, an adaptive scheme which controls the time-varying values of P and a_j is presented and tested.

A new high-order local Absorbing Boundary Condition (ABC) has been recently proposed for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. It is based on the stress–velocity formulation of the elastodynamics problem, and on the general Complete Radiation Boundary Condition (CRBC) approach, originally devised by Hagstrom and Warburton in 2009. The work presented here is a sequel to previous work that concentrated on the stability of the scheme; this is the first known high-order ABC for elastodynamics which is long-time stable. Stability was established both theoretically and numerically. The present paper focuses on the accuracy of the scheme. In particular, two accuracy-related issues are investigated. First, the reflection coefficients associated with the new CRBC for different types of incident and reflected elastic waves are analyzed. Second, various choices of computational parameters for the CRBC, and their effect on the accuracy, are discussed. These choices include the optimal coefficients proposed by Hagstrom and Warburton for the acoustic case, and a simplified formula for these coefficients. A finite difference discretization is employed in space and time. Numerical examples are used to experiment with the scheme and demonstrate the above-mentioned accuracy issues.

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space $H_0^m(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is any bounded, smooth, open subset of ${ℝ}^{2m}$, $m\ge 1$. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

To improve the performance of smoke recognition, it is important to extract representative and discriminative features for smoke. We propose a learning based method to automatically extract discriminative features from raw pixels. First, patch-based local differences rather than single-pixel-based differences are computed. Then a sampling strategy is learnt to extract dominant local differences. Third, the sampled differences are used to learn a codebook, through which differences are projected into a discriminative feature space. Thus, feature matrices consisting of discriminative differences can be obtained according to the codebook. At last, the feature maps are encoded without quantization, thus final feature is computed. Experimental results demonstrate that the proposed method achieves better performance than some existing handcrafted and learning based feature extraction methods.

A compact high-order scheme has been successfully proposed and verified in this paper. In this scheme, the traditional gradient reconstruction was replaced with a compact scheme. There were no needs to modify the process and algorithms of unstructured FVM including boundary conditions, flux technique, limiter functions and so on. Both memory and computation loads with the new scheme were not increased than the traditional one. Additionally, we modified Venkatakrishnan limiter to suppress numerical oscillation. The proposed compact scheme and modified Venkatakrishnan limiter have been verified with numerical experiments on benchmark problems. Numerical results showed good agreement with those obtained by other methods.

In this study, we present a fourth-order and a sixth-order blended compact difference (BCD) schemes for approximating the three-dimensional (3D) convection–diffusion equation with variable convective coefficients. The proposed schemes, where transport variable, its first and second derivatives are carried as the unknowns, combine virtues of compact discretization, fourth-order Padé scheme and sixth-order combined compact difference (CCD) scheme for spatial derivatives and can efficiently capture numerical solutions of linear and nonlinear convection–diffusion equations with Dirichlet boundary conditions. The fourth-order scheme requires only 7 grid points and the sixth-order scheme requires 19 grid points. The distinguishing feature of the present method is that methodologies of explicit compact difference and implicit compact difference are blended together. The truncation errors of the two difference schemes are analyzed for the interior grid points, respectively. Simultaneously, a sixth-order accuracy scheme is proposed to compute the first and second derivatives of the grid points on boundaries. Finally, the presented methods are applied to several test problems from the literature including linear and nonlinear problems. It is found that the presented schemes exhibit good performance.

In this paper we consider a generalized n-th order delay differential equation, by applying Mawhin's continuation theory and some new inequalities, we obtain sufficient conditions for the existence of periodic solutions. Moreover, an example is given to illustrate the results.