On highly efficient derivative-free family of numerical methods for solving polynomial equation simultaneously

Although solving polynomial equations is important in mathematics, most high school students can solve only linear and quadratic equations. This is because the methods for solving cubic and quartic equations are difficult, and no general methods of solution are available for equations of degree higher than four. However, numerical methods can be used to approximate the real solutions of polynomial equations of any degree. Because they involve a great deal of computation they have not traditionally been taught in the schools. Now that most students have access to calculators and computers, this computational difficulty is easily overcome.

Numerical simulation of three-dimensional (3D) seismic wavefields forms the basis of the research on the migration methods of 3D seismic data based on wave equations. Because the simulation precision of wavefield extrapolation determines the imaging accuracy to a certain extent, it is very important to study how to enhance the forward modeling precision of 3D seismic wavefields. Thus, we build on an optimized 3D staggered-grid finite-difference (SFD) method with high simulation precision based on two-dimensional (2D) seismic modeling. Since it generates the corresponding difference coefficients by utilizing the least square (LS) method to minimize the objective function constructed by the time-space domain dispersion relation of the 3D acoustic wave equation, our optimized time-space domain LS-based 3D SFD method can effectively enhance the modeling precision of the 3D seismic wavefields in theory compared with the 3D SFD methods based on the Taylor-series expansion (TE), especially for the large wavenumber range. Examining the numerical dispersion, algorithm stability and computational cost, we compare our optimized time-space domain LS-based 3D SFD method with three conventional TE-based and LS-based 3D SFD methods to illustrate and demonstrate its effectiveness and feasibility. The numerical examples from different 3D models suggest that our optimized time-space domain LS-based 3D SFD method can generate less numerical dispersion and higher simulation accuracy for 3D seismic wavefields than three other conventional 3D SFD methods, but its stability condition is stricter and its computational cost is slightly higher.

The error control of local interpolation for a 2D MLFMA will be discussed. The way to select proper parameters is proposed in terms of both numerical accuracy and computational cost. Satisfying the conditions derived in this paper, error can be controlled at the same level as global interpolation, and the computational cost becomes less expensive than the global one. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 36: 8–12, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10655

Nearly analytic discrete method (NADM) is a higher accurate method for elastic wave equation that can suppress effectively numerical dispersion caused by discretizing the wave equation. In this paper we investigate the efficient implementation of NADM and present a refinement of the original NADM. Our theoretical analyses show that the modified NADM can improve significantly over the original one in numerous perspectives including numerical errors, storage spaces, and computational costs. Three-component synthetic VSP seismograms in 3-layered transversely isotropic (TI) media generated by the modified NADM are also reported. Theoretical analyses and numerical results show that the modified NADM can reduce storage space about 53 percent and computational costs about 30 percent compared with the original NADM. Moreover the accuracy of the modified NADM in time increases from 2-order of the original NADM to 4-order. Numerical results suggest that the modified NADM is more suitable to large-scale modeling because the modified method has little numerical dispersions even when too-coarse grids are used.

A Comparative Study of Locally Conservative Numerical Methods for Darcy's Flows

It is important to estimate physical properties of cracked rocks in exploration geophysics. In the past, many analytical studies for elastic properties of a cracked medium have been conducted, for example non-interacting theory, self-consistent theory and so on. However, some limitations (e.g. crack geometries, long wavelength limit) always remain in theoretical approaches. On the other hand, numerical studies, which are much less restrictive, have also been conducted using several numerical methods. In the present study, we apply a Hamiltonian particle method (HPM) for estimating effective elastic properties of cracked media because HPM has some advantages in treating cracked media. In order to investigate the validity of the particle method for determining the effective properties of cracked media, we calculate the effective velocity of cracked region using the particle method and compare with those from several effective medium theories. Our results show good agreements with those from modified self-consistent theory. Since this is consistent with the previous studies, HPM can estimate effective elastic properties of cracked media. HPM has some advantages about computational costs, this indicates that the effective calculation with low computation costs (numerical memory and calculation costs) is possible.

This paper investigates the prediction of the depth of penetration (DOP) for concrete targets under high-speed projectile impact using multiple simulation algorithms in LS-DYNA. Three numerical methods, i.e., the traditional finite element method (FEM), a fixed-coupling FEM-SPH (Smooth Particle Hydrodynamics) model, and an adaptive coupling FEM-SPH model, are employed to simulate the penetration processes. The computational results are compared against established empirical formulas to evaluate their predictive accuracy and efficiency. The findings indicate a distinct trade-off between numerical precision and computational cost. The adaptive FEM-SPH algorithm achieves the highest accuracy, with a maximum error of less than 10% across considered velocity ranges, and effectively captures cavity expansion effects. The standard FEM algorithm offers the highest computational efficiency, requiring less than half the time of the other methods, albeit with a maximum error of up to 25%. The fixed-coupling FEM-SPH model provides an intermediate solution, showing improved accuracy at velocities above 400 m/s but lower efficiency. This comparative analysis offers a practical guideline for selecting appropriate simulation techniques in protective structure design, balancing the demands for rapid estimation, detailed physical insight, and final safety verification.

High-order, stable, and efficient pseudospectral method using barycentric Gegenbauer quadratures

A high-throughput hybrid task and data parallel Poisson solver for large-scale simulations of incompressible turbulent flows on distributed GPUs

Chapter 1 - The characteristic set method and its applications

It is known that homotopy continuation methods (HCM) used in con- junction with a classical numerical method to compute roots of nonlinear algebraic equations can improve the performance of the classical numerical method. In this pa- per, we develop an Ostrowski-HCM method and apply it to solve several polynomial equations. The results obtained indicate that Ostrowski-HCM performs better than Ostrowski’s method. Keywords : Numerical method; Polynomial equations; Ostrowski’s method; Homo- topy continuation method. 2010 Mathematics Subject Classification 65H05; 65H10; 65H20.

The numerical simulation of the tempered fractional Black–Scholes equation for European double barrier option

A model for the time-domain scattered intensity from a heterogeneous layered seafloor due to a point source has been recently developed by Tang and Jackson [J. Acoust. Soc. Am. Suppl. 4, 140, 3363]. To accurately model measurements taken in the deep ocean with source / receiver altitudes on the order of 5 km, the computational cost of this model is quite high, with computational cost scaling as O((kH)^2), where k is the acoustic wavenumber, and H is the source / receiver altitude. We present a fast method of numerical integration based on the work of Levin that greatly reduces the computational cost for deep water scenarios. In the absence of a complex resonant structure in the sediment, the proposed numerical integration scheme reduces the scaling to O(1). If this resonant structure is present, then the scaling is approximately O((kD)^2), where D is the total sediment thickness. Model results from deep ocean environments, including turbidite seafloors are presented and discussed.

Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of Bézout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications.

On the use of the Discrete-Time Convolution Model for the compensation of non-idealities within digital acquisition channels