Direct expansion of Fourier extrapolator for one-way wave equation using Chebyshev polynomials of the second kind

The Fourier method for one-way wave propagation is efficient, but potentially inaccurate in complex media. The implicit finite-difference method can handle arbitrarily complex media, but can be inefficient in 3D and has limited dip bandwidth. We proposed a new Fourier method based on Chebyshev expansion of the second kind. Both theoretical analyses and numerical experiments show that the proposed method is comprehensively superior to a similar method based on Chebyshev expansion of the first kind in terms of balanced amplitude and error tolerance. Within the dip bandwidth from 0 to 65°, the fourth-order form of our method has an error tolerance of 2%, which is about one-third that of Chebyshev expansion of the first kind. Our method is also superior to the implicit finite-difference method in several important aspects: effective bandwidth, computational efficiency, numerical dispersion and two-way splitting error. It can be easily extended from 2D to 3D compared with the finite-difference method and from low orders to high orders compared with the optimized Chebyshev-Fourier method. The proposed method shows better imaging results of the SEG/EAGE model by providing a well-focused salt dome, flank and bottom as well as the detailed structures beneath the salt body, compared with the implicit finite-difference method and Chebyshev expansion of the first kind; meanwhile, our method has less imaging artifacts since it can better position the reflectors.

Simulasi Numerik Model Matematika Arus Lalu Lintas Berbasis Fungsi Velositas Underwood

Abstrak.Model matematika arus lalu lintas pertama kali dikembangkan oleh Lighthill, Whitham dan Richards pada tahun 1956 yang dikenal dengan model (LWR). Dalam model LWR, fungsi kecepatan adalah unsur yang terpenting. Dalam makalah ini digunakan fungsi kecepatan underwood karena memiliki tingkat kesesuaian yang terbaik dibadingkan dengan fungsi kecepatan lainnya. Metode beda hingga implisit digunakan untuk menemukan solusi numerik model LWR dengan model kecepatan Underwood. Konvergensi metode beda hingga implisit dibuktikan dengan menggunakan teorema Ekuivalensi Lax. Simulasi numerik jalan raya satu lajur sepanjang 10 km dilakukan selama 1 jam menggunakan metode beda hingga implisit berdasarkan data awal dan batas yang dibuat secara artifisial. Simulasi numerik dilakukan dengan dua parameter berbeda. Hasil eksperimen menujukkan bahwa semakin tinggi rata-rata kepadatan kendaraan pada suatu laju mengakibatkan rata-rata kecepatan kendaraan akan berkurang. Kata kunci: Metode Beda Hingga Implisit, Model LWR, Arus Lalu Lintas, Fungsi Felositas Underwood, Simulasi Numerik.Kata kunci : Abstract. Mathematical traffic flow model was first developed by Lighthill, Whitham and Richards in 1956, known as (LWR) model. In LWR model, velocity function was most important. In this paper, Underwood velocity function was used. Implicit finite difference method used to found the numerical solution of LWR model with Underwood velocity model. Convergence the implicit finite difference method proved using the Lax equivalence theorem. The numerical simulation of 10 km highway of single lane was performed for 1 hours using the implicit finite difference method based on artificially generated initial and boundary data. Numerical simulation performed with two different parameters. An experimental result for the stability condition of the numerical scheme was also presented. Density, velocity, and fluks for 1 hours was experimental result of numerical simulation.Keywords: Implicit finite difference method, Lax equivalence theorem, LWR model, Traffic flow, Under-wood velocity Function, Numerical simulation.

A Front-Fixing Implicit Finite Difference Method for the American Put Options Model

In this paper, we present an implicit finite difference method for the numerical solution of the Black–Scholes model of American put options without dividend payments. We combine the proposed numerical method by using a front-fixing approach where the option price and the early exercise boundary are computed simultaneously. We study the consistency and prove the stability of the implicit method by fixing the values of the free boundary and of its first derivative. We improve the accuracy of the computed solution via a mesh refinement based on Richardson’s extrapolation. Comparisons with some proposed methods for the American options problem are carried out to validate the obtained numerical results and to show the efficiency of the proposed numerical method. Finally, by using an a posteriori error estimator, we find a suitable computational grid requiring that the computed solution verifies a prefixed error tolerance.

A Study of Photo-Thermoelastic Wave in Semiconductor Materials With Spherical Holes Using Analytical-numerical Methods

Analytical and numerical solutions are two basic tools in the study of photothermal interaction problems in semiconductor medium. In this paper, we compare the analytical solutions with the numerical solutions for thermal interaction in semiconductor mediums containing spherical cavities. The governing equations are given in the domain of Laplace transforms and the eigenvalues approaches are used to obtained the analytical solution. The numerical solutions are obtained by applying the implicit finite difference method (IFDM). A comparison between the numerical solutions and analytical solution are presented. It is found that the implicit finite difference method (IFDM) is applicable, simple and efficient for such problems.

Numerical Investigation of the Unsteady Flows in Hydropower Plants

One of the most important hazards that threatens the stability of power plant buildings is the phenomenon of water hammer, which can occur in the Penstock pipe of a turbine due to the rapid opening and closing of a valve. Fluid Descriptive Equations in this situation, there are two hyperbolic partial nonlinear partial differential equations that are very difficult and complex to solve analytically and are possible only for very simple conditions. In this study, by examining the two numerical methods of characteristic lines and implicit finite difference with Verwy & Yu schema, which are widely used in the analysis of instabilities, their disadvantages and advantages are clearly clarified and a suitable comparison basis for use. They should be provided in different conditions in hydropower plant. The results of the characteristic method in terms of maximum and minimum pressure show more and less values than the implicit finite difference method. In the characteristic method, perturbations and fast wave fronts are presented with more accuracy than the implicit finite difference method. At points near the upstream, downstream and middle boundaries, the accuracy of the characteristic method in presenting pressure and flow fluctuations is higher than the implicit finite difference method. In the characteristic method, it is recommended not to use certain time steps and try as much as possible avoid interpolation by selecting the appropriate time step. The results of examining the amount of changes in coefficient of friction in both methods show that it is not correct to take its value constant (proportional to the value obtained in stable conditions) and coefficient of friction should be calculated in proportion to changes in velocity at different times and used in the governing equation.

Variable-order optimal implicit finite-difference schemes for explicit time-marching solutions to wave equations

The time-space-domain finite-difference method is widely used in forward modeling of wave equations. Conventional explicit finite-difference methods (with high-order in space and the second-order in time) would result in apparent temporal and spatial dispersion for high frequencies and large time steps. Moreover, the saturation effect of Taylor expansion seriously restricts the improvement of bandwidth coverage and efficiency of the explicit finite-difference methods. We proposed a variable-order optimization scheme of the implicit finite-difference method to improve the computational efficiency and numerical accuracy of forward modeling. Then, we applied time-dispersion transforms to filter out the temporal dispersion generated by the second-order temporal approximations. The proposed method greatly alleviates the saturation effect of the high-order spatial finite-difference operators. Dispersion analysis shows that the optimized coefficients of the implicit finite-difference method based on the Remez algorithm can achieve the widest bandwidth (close to the Nyquist wavenumber), which corresponds to the shortest length of the spatial finite-difference operator under a given error threshold. The proposed method has great potential to approach the highest spectral accuracy but with minimal increase in computational cost. Numerical experiments show that the combination of the variable-order optimization of implicit finite-difference method and the time-dispersion transforms can significantly reduce numerical dispersion. Compared with traditional methods, our scheme is more advantageous for numerical simulation of large-scale geological models, since it has the least amount of calculation burden under the same accuracy requirements.