A septic B-spline collocation method for solving the generalized equal width wave equation

In this work, a septic B-spline collocation method is implemented to find the numerical solution of the generalized equal width (GEW) wave equation by using two different linearization techniques. Test problems including single soliton, interaction of solitons and Maxwellian initial condition are solved to verify the proposed method by calculating the error norms L2 and L∞ and the invariants I1, I2 and I3. Applying the Von-Neumann stability analysis, the proposed method is shown to be unconditionally stable. As a result, the obtained results are found in good agreement with the some recent results

Application of the collocation method with b-splines to the gew equation

In this paper, the generalized equal width (GEW) wave equation is solved numerically by using a quintic B-spline collocation algorithm with two different linearization techniques. Also, a linear stability analysis of the numerical scheme based on the von Neumann method is investigated. The numerical algorithm is applied to three test problems consisting of a single solitary wave, the interaction of two solitary waves, and a Maxwellian initial condition. In order to determine the performance of the numerical method, we compute the error in the L2- and L∞ norms and in the invariants I1, I2, and I3 of the GEW equation. These calculations are compared with earlier studies. Afterwards, the motion of solitary waves according to different parameters is designe

Numerical investigations of shallow water waves via generalized equal width (GEW) equation

In this article, a mathematical model representing solution of the nonlinear generalized equal width (GEW) equation has been considered. Here we aim to investigate solutions of GEW equation using a numerical scheme by using sextic B-spline Subdomain finite element method. At first Galerkin finite element method is proposed and a priori bound has been established. Then a semi-discrete and a Crank-Nicolson Galerkin finite element approximation have been studied respectively. In addition to that a powerful Fourier series analysis has been performed and indicated that our method is unconditionally stable. Finally, proficiency and practicality of the method have been demonstrated by illustrating it on two important problems of the GEW equation including propagation of single solitons and collision of double solitary waves. The performance of the numerical algorithm has been demonstrated for the motion of single soliton by computing L∞ and L2 norms and for the other problem computing three invariant quantities I1, I2 and I3. The presented numerical algorithm has been compared with other established schemes and it is observed that the presented scheme is shown to be effectual and valid

Numerical solutions of the generalized equal width wave equation using the Petrov–Galerkin method

In this article, we consider a generalized equal width wave (GEW) equation which is a significant nonlinear wave equation as it can be used to model many problems occurring in applied sciences. Here we study a Petrov–Galerkin method for the model problem, in which element shape functions are quadratic and weight functions are linear B-splines. We investigate the existence and uniqueness of solutions of the weak form of the equation. Then, we establish the theoretical bound of the error in the semi-discrete spatial scheme as well as of a full discrete scheme at t = t n. Furthermore, a powerful Fourier analysis has been applied to show that the proposed scheme is unconditionally stable. Finally, propagation of solitary waves and evolution of solitons are analyzed to demonstrate the efficiency and applicability of the proposed scheme. The three invariants (I1, I2 and I3) of motion have been commented to verify the conservation features of the proposed algorithms. Our proposed numerical scheme has been compared with other published schemes and demonstrated to be valid, effective and it outperforms the others

Spatiotemporal properties of multiscale two-dimensional ows

The extraordinary complexity of turbulence has motivated the study of some of its key features in flows with similar structure but simpler or even trivial dynamics. Recently, a novel class of such flows has been developed in the laboratory by applying multiscale electromagnetic forcing to a thin layer of conducting fluid. In spite of being stationary, planar, and laminar these flows have been shown to resemble turbulent ones in terms of energy spectra and particle dispersion. In this thesis, some extensions of these flows are investigated through simulations of a layer-averaged model carried out using a bespoke semi-Lagrangian spline code. The selected forcings generalise the experimental ones by allowing for various kinds of self-similarity and planetary motion of the multiple scales. The spatiotemporal structure of the forcings is largely reflected on the flows, since they mainly arise from a linear balance between forcing and bottom friction. The exponents of the approximate power laws found in the wavenumber spectra can thus be related to the scaling and geometrical forcing parameters. The Eulerian frequency spectra of the unsteady flows exhibit similar power laws originating from the sweeping of the multiple flow scales by the forcing motions. The disparity between fluid and sweeping velocities makes it possible to justify likewise the observed Lagrangian power laws, but precludes a proper analogy with turbulence. In the steady case, the absolute dispersion of tracer particles presents ballistic and diffusive stages, while relative dispersion shows a superquadratic intermediate stage dominated by separation bursts due to the various scales. In the unsteady case, the absence of trapping by fixed streamlines leads to appreciable enhancement of relative dispersion at low and moderate rotation frequency. However, the periodic reversals of the large scale give rise to subdiffusive absolute dispersion and severely impede relative dispersion at high frequency

Temperature reconstruction and acoustic time of flight determination for boiler furnace exit temperature measurement

The furnace exit gas temperature (FEGT) is one of the fundamental parameters necessary to determine the energy balance of the boiler in a coal-fired power plant, and is thus beneficial to the production of reliable thermo-fluid models of its operation and the operation of the systems down and upstream. The continuous measurement of the FEGT would also be a useful indicator to predict, prevent and diagnose faults, optimize boiler operation and aid the design of heat transfer surfaces. Acoustic pyrometry, a technique that measures temperature based on the travel time of an acoustic wave in a gas, is investigated as a viable solution for continuous direct measurement of the FEGT. This study focuses specifically on using acoustic pyrometry to reconstruct the temperature profile at the furnace exit including methods for accurately determining the time of flight (TOF) of acoustic waves. An improved reconstruction technique using radial basis functions (RBF) for interpolation and a least squares algorithm is simulated and its performance was compared to cubic spline interpolation, regression and Lagrange interpolation by evaluating its reconstruction accuracy in terms of mean and root-mean-squared (RMS) error when reconstructing set temperature profiles. Various parameters including transceiver positions, grid divisions and time of flight error, are investigated in terms of how they inform acoustic pyrometry implementation. The improved RBF interpolation function managed to reconstruct complex temperature profiles and had a greater reconstruction accuracy than compared interpolation methods, improving on the accuracy of previous work done. Random acoustic path error was found to not be additive with reconstruction error however repeating acoustic TOF readings improved reconstruction accuracy to mitigate this effect. In general, it was also found that symmetrical transmitter/receiver positions produced more accurate reconstructions as well as positioning receivers/transceivers and grid lines closer to the furnace walls, where the greatest temperature change occurs. In addition to testing reconstruction methods, a low-cost experimental set-up was constructed to measure the time of flight. The focus of this study was on using various signal processing methods to determine the time of flight and evaluating their accuracy in the presence of noise. Methods such as threshold detection with bandpass filtering, cross correlation, generalized cross-correlation (GCC) and a new method developed employing variable notch filters with locations and widths based on repetitive frequencies identified in the noise with cross correlation. The performance of methods was experimentally tested under varying signal to noise ratios (SNR) and noise conditions. These SNR tests showed that cross-correlation methods produced more reliable TOF readings under lower SNRs than threshold detection methods. Under white noise the smooth coherent transform (SCOT) GCC variation proved to produce the most accurate results producing an average TOF error of 0.84 % up until a SNR of 1.4 before reducing in accuracy. In coloured noise (generated based on previous boiler recordings) the variable notch filter method with crosscorrelation was able to identify repetitive noise frequencies filter them out and ultimately produced results with an average TOF error of 1.99 % up until a SNR of 0.67, where the noise level exceeds that of the signal

A decision-making machine learning approach in Hermite spectral approximations of partial differential equations

The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains, are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decision-making process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications.Comment: 22 pages, 4 figure

GEW ve GRLW denklemlerinin sonlu elemanlar yöntemi ile sayisal çözümleri

Bu tez c¸alıs¸masında, GEW ve GRLW denklemleri, B-spline fonksiyonlar kullanılarak kollokasyon ve Galerkin sonlu elemanlar yontemleri ile sayısal olarak çozüldü. Von-Neumann tekniği kullanılarak, lineerleştirilmis¸ algoritmaların şartsız kararlı olduğu g österildi. Sayısal algoritmalar; tek solitary dalga, iki ve üç¸ solitary dalganın etkileşimi, Maxwellian başlangıç şartı ile dalga oluşumu ve ardışık dalgaların gelişimini içeren orneklere uygulanarak test edildi. Sayısal algoritmaların performansını kanıtlamak için, L2 ve L∞ hata normları hesaplandı ve daha önce elde edilen sayısal sonuçlarla karşılaştırıldı. Sayısal algoritmaların kütle, momentum ve enerji ile ilgili ozellikleri koruduğunu göstermek için I1, I2 ve I3 ile ifade edilen korunum sabitlerindeki degişim hesaplandı. Ayrıca, solitary dalgaların farklı zamanlardaki hareketleri grafik çizilerek gosterildi

Isogeometric analysis: an overview and computer implementation aspects

Isogeometric analysis (IGA) represents a recently developed technology in computational mechanics that offers the possibility of integrating methods for analysis and Computer Aided Design (CAD) into a single, unified process. The implications to practical engineering design scenarios are profound, since the time taken from design to analysis is greatly reduced, leading to dramatic gains in efficiency. The tight coupling of CAD and analysis within IGA requires knowledge from both fields and it is one of the goals of the present paper to outline much of the commonly used notation. In this manuscript, through a clear and simple Matlab implementation, we present an introduction to IGA applied to the Finite Element (FE) method and related computer implementation aspects. Furthermore, implemen- tation of the extended IGA which incorporates enrichment functions through the partition of unity method (PUM) is also presented, where several examples for both two-dimensional and three-dimensional fracture are illustrated. The open source Matlab code which accompanies the present paper can be applied to one, two and three-dimensional problems for linear elasticity, linear elastic fracture mechanics, structural mechanics (beams/plates/shells including large displacements and rotations) and Poisson problems with or without enrichment. The Bezier extraction concept that allows FE analysis to be performed efficiently on T-spline geometries is also incorporated. The article includes a summary of recent trends and developments within the field of IGA