High-order Finite Difference Technique Research Articles

Sonic boom reflection over an isolated building and multiple buildings.

Sonic boom reflection is investigated over an isolated building and multiple buildings using numerical simulations. For that, the two-dimensional Euler equations are solved using high-order finite-difference techniques. Three urban geometries are considered for two boom waves, a classical N-wave and a low-boom wave. First, the variations of the pressure waveforms and the corresponding perceived noise are analyzed along an isolated building. The influence of the building is limited to an illuminated region at its front and a shadow region at its rear, whose size depends on the building's height and the Mach number. Two buildings are then considered. In addition to arrivals related to reflection on the building facades or to diffraction at the building corners, low-frequency oscillations, associated with resonances, are noticed in the street canyon. Their amplitude depends on the street width and on the incident boom frequency contents. Despite their significance, these low-frequency oscillations have little impact on the perceived noise. Finally, a periodic distribution of identical buildings is examined. The duration of the waveforms is notably increased due to multiple diffraction and canyon resonances. Variations in perceived noise at ground level are moderate for large streets, but become noticeable as the street width reduces.

Supervised machine learning of ultracold atoms with speckle disorder

We analyze how accurately supervised machine learning techniques can predict the lowest energy levels of one-dimensional noninteracting ultracold atoms subject to the correlated disorder due to an optical speckle field. Deep neural networks with different numbers of hidden layers and neurons per layer are trained on large sets of instances of the speckle field, whose energy levels have been preventively determined via a high-order finite difference technique. The Fourier components of the speckle field are used as the feature vector to represent the speckle-field instances. A comprehensive analysis of the details that determine the possible success of supervised machine learning tasks, namely the depth and the width of the neural network, the size of the training set, and the magnitude of the regularization parameter, is presented. It is found that ground state energies of previously unseen instances can be predicted with an essentially negligible error given a computationally feasible number of training instances. First and second excited state energies can be predicted too, albeit with slightly lower accuracy and using more layers of hidden neurons. We also find that a three-layer neural network is remarkably resilient to Gaussian noise added to the training-set data (up to 10% noise level), suggesting that cold-atom quantum simulators could be used to train artificial neural networks.

High-order finite difference technique for delay pseudo-parabolic equations

One dimensional initial–boundary delay pseudo-parabolic problem is being considered. To solve this problem numerically, we construct higher order difference method for approximation to the considered problem and obtain the error estimate for its solution. Based on the method of energy estimate the fully discrete scheme is shown to be convergent of order four in space and of order two in time. Numerical example is presented.

Seeding of Görtler vortices through a suction and blowing strip

The resulting wavelength of Gortler vortices in boundary layers over concave surfaces is determined by the upstream history of the flow and by wall disturbances such as roughness, heating/cooling strips or suction and blowing. In isotropic disturbance conditions, the predominant wise wavelength corresponds to the strongest growing vortex mode predicted by the linear stability theory. If the disturbance environment is not isotropic, vortices with wavelength different from the one with the highest growth rate may emerge. The present investigation considers the wavelength selection when Gortler vortices are excited by a suction and blowing strip at the wall. The study is based on numerical simulations of the vorticity transport equations derived from the Navier-Stokes equations. They are solved using a compact high-order finite difference technique. The results show that, when the vortices are excited by suction and blowing at the wall, their wise wavelength does not necessarily correspond to the imposed wavelength. Curves of streamwise development of the disturbance energy for different harmonics are presented, showing the evolution of the dominant modes. Isolines of streamwise velocity in the wise plane are also presented, showing how the higher harmonics distort the characteristic mushroom structures.

High-order finite-difference techniques for linear singular perturbation boundary value problems

Three high-order finite-difference techniques are developed for the second-order, singularity perturbed linear boundary value problem in one dimension. Taylor series expansions and error conversions are used for the development of the techniques. Convergence and stability conditions of these techniques are proved, and some numerical results are given.

High-Order "Cyclo-Difference" Techniques: An Alternative to Finite Differences

The summation-by-parts energy norm is used to establish a new class of high-order finite-difference techniques referred to here as "cyclo-difference" techniques. These techniques are constructed cyclically from stable subelements and require no special numerical procedures near the boundaries; when coupled with the simultaneous approximation term (SAT) boundary treatment, they are time asymptotically stable for an arbitrary hyperbolic system. These techniques are similar to spectral multi-domain techniques and are ideally suited for parallel implementation, but they do not require special collocation points. The principal focus of this work is on methods of sixth-order formal accuracy or less; however, these methods could be extended in principle to any arbitrary order of accuracy.

A boundary value approach for estuarine water quality modelling with results for Jamaica Bay, New York

Results of water quality modelling for Jamaica Bay, a New York estuary with a large hydraulic circulation, are presented. The two-dimensional topology is approximated by a set of coupled one-dimensional subsystems. The long term steady state water quality problem is then reformulated as a multi-point boundary value problem for ordinary differential equations. Piecewise constant dispersion parameters are estimated from salinity data. A sequential algorithm based on parallel shooting is developed for solving the multipoint problem. The method, which simplifies handling of feedforward and feedback reaction kinetics, is equivalent to employing a high order finite difference technique with the subsequent enhanced accuracy. Results of model verification for uncoupled variables — salinity, coliform, total soluble phosphorus, and coupled variables for nitrogen (organic and ammonia) and BOD-DO for Jamaica Bay are discussed.