First-order accuracy methods with conformable stability regions

2000 ◽  
Vol 15 (2) ◽  
pp. 183-192
Author(s):  
E. A. NOVIKOV ◽  
L. N. KONTARYOVA
Keyword(s):  
Order Accuracy ◽  
Stability Regions ◽  
First Order

scholarly journals On the integration processes of A. Huťa

1963 ◽  
Vol 3 (2) ◽  
pp. 202-206 ◽  
Author(s):  
J. C. Butcher
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Sixth Order ◽  
Order Accuracy ◽  
First Order ◽  
Runge Kutta ◽  
Image Position

Huta [1], [2] has given two processes for solving a first order differential equation to sixth order accuracy. His methods are each eight stage Runge-Kutta processes and differ mainly in that the later process has simpler coefficients occurring in it.

A CANADIAN GRID SYSTEM IN 3° TRANSVERSE MERCATOR ZONES

1964 ◽  
Vol 18 (2) ◽  
pp. 147-155
Author(s):  
J. Saastamoinen
Keyword(s):  
Free Choice ◽  
Metropolitan Areas ◽  
Central Meridian ◽  
Conformal Map ◽  
Grid System ◽  
Order Accuracy ◽  
Map Projection ◽  
First Order ◽  
The Subject ◽  
Local Plane

All geodetic networks that rest previously fixed control are best computed in plane coordinates. Under this category fall a great deal of first-order triangulation and, of course, the whole volume of lower-order work—all the way down to the last monuments from which detail surveys originate. A geodetic grid must be designed to meet first-order accuracy and should be established on federal rather than provincial level. Its use requires precomputed tables based on some conformal map projection, preferably the Transverse Mercator (Gauss-Krüger) projection. Congruent projection zones, simple scale factor, free choice of central meridian for local plane coordinates in cities and metropolitan areas—all these features of the Transverse Mercator find no parallel in any other map projection. Following a brief introduction to the subject, a set of tables for a proposed Canadian grid system is presented.

scholarly journals Unsplit complex frequency-shifted PML implementation using auxiliary differential equations for seismic wave modeling

Geophysics ◽  
2010 ◽  
Vol 75 (4) ◽  
pp. T141-T154 ◽  
Author(s):  
Wei Zhang ◽  
Yang Shen
Keyword(s):  
Free Surface ◽  
Finite Difference ◽  
Differential Equations ◽  
Seismic Wave ◽  
Numerical Scheme ◽  
Complex Frequency ◽  
Order Accuracy ◽  
First Order ◽  
Time Marching ◽  
Interior Domain

The complex-frequency-shifted perfectly matched layer (CFS-PML) technique can efficiently absorb near-grazing incident waves. In seismic wave modeling, CFS-PML has been implemented by the first-order-accuracy convolutional PML technique or second-order-accuracy recursive convolution PML technique. Both use different algorithms than the numerical scheme for the interior domain to update auxiliary memory variables in the PML and thus cannot be used directly with higher-order time-marching schemes. We work with an unsplit-field CFS-PML implementation using auxiliary differential equations (ADEs) to update the auxiliary memory variables. This ADE CFS-PML results in complete first-order differential equations. Thus, the numerical scheme for the interior domain can be used to solve ADE CFS-PML equations. We have implemented ADE CFS-PML in the finite-difference time-domain method and in anonstaggered-grid finite-difference method with the fourth-order Runge-Kutta scheme, demonstrating its straightforward implementation in different numerical time-marching schemes. We have also theoretically analyzed the role of the scalingfactor of CFS-PML; it transforms the PML to a transversely isotropic material, reducing the effective wave speed normal to the PML layer and bending the wavefront toward the normal direction of the PML layer. Our numerical tests indicate that the optimal value reduces the points per dominant wavelength at the outermost boundary to three, about half the value required by the numerical scheme. We also have found that the PML equations should be derived taking the free-surface boundary condition into account in finite-difference methods. Otherwise, the free surface in the PML layer causes instability or ineffective absorption of surface waves. Tests show that we can use a narrow-slice mesh with ADE CFS-PML to simulate full wave propagation efficiently in models with complex structure.

scholarly journals Algorithms of Finite Difference for Pricing American Options under Fractional Diffusion Models

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Jun Xi ◽  
Yanqing Chen ◽  
Jianwen Cao
Keyword(s):  
Differential Equation ◽  
American Options ◽  
Compound Poisson Process ◽  
Fractional Diffusion ◽  
Diffusion Models ◽  
Singularity Problem ◽  
Order Accuracy ◽  
Fractional Difference ◽  
Finite Moment ◽  
First Order

It is well known that linear complementarity problem (LCP) involving partial integro differential equation (PIDE) arises from pricing American options under Lévy Models. In the case of infinite activity process, the integral part of the PIDE has a singularity, which is generally approximated by a small Brownian component plus a compound Poisson process, in the neighborhood of origin. The PIDE can be reformulated as a fractional partial differential equation (FPDE) under fractional diffusion models, including FMLS (finite moment log stable), CGMY (Carr-Madan-Geman-Yor), and KoBol (Koponen-Boyarchenko-Levendorskii). In this paper, we first present a stable iterative algorithm, which is based on the fractional difference approach and penalty method, to avoid the singularity problem and obtain numerical approximations of first-order accuracy. Then, on the basis of the first-order accurate algorithm, spatial extrapolation is employed to obtain second-order accurate numerical estimates. Numerical tests are performed to demonstrate the effectiveness of the algorithm and the extrapolation method. We believe that this can be used as necessary tools by the engineers in research.

First-order ray tracing for qP waves in inhomogeneous, weakly anisotropic media

Geophysics ◽  
2005 ◽  
Vol 70 (6) ◽  
pp. D65-D75 ◽  
Author(s):  
Ivan Pšenčík ◽  
Véronique Farra
Keyword(s):  
Ray Tracing ◽  
Anisotropic Media ◽  
Order Accuracy ◽  
Weak Anisotropy ◽  
Higher Symmetry ◽  
First Order ◽  
Isotropic Media ◽  
Anisotropy Parameters ◽  
Order Quantities ◽  
Relative Errors

We propose approximate ray-tracing equations for qP-waves propagating in smooth, inhomogeneous, weakly anisotropic media. For their derivation, we use perturbation theory, in which deviations of anisotropy from isotropy are considered to be the first-order quantities. The proposed ray-tracing equations and corresponding traveltimes are of the first order. Accuracy of the traveltimes can be increased by calculating a secondorder correction along first-order rays. The first-order ray-tracing equations for qP-waves propagating in a general weakly anisotropic medium depend on only 15 weak-anisotropy parameters (generalization of Thomsen’s parameters). The equations are thus considerably simpler than the exact ray-tracing equations. For higher-symmetry anisotropic media the equations differ only slightly from equations for isotropic media. They can thus substitute for the traditional isotropic ray tracers used in seismic processing. For vanishing anisotropy, the first-order ray-tracing equations reduce to standard, exact ray-tracing equations for isotropic media. Numerical tests for configuration and models used in seismic prospecting indicate negligible dependence of accuracy of calculated traveltimes on inhomogeneity of the medium. For anisotropy of about 8%, considered in the examples presented, the relative errors of the traveltimes, including the second-order correction, are well under 0.05%; for anisotropy of about 20%, they do not exceed 0.3%.

A Numerical Solution of Three-Dimensional Problems in Dynamic Elasticity

1970 ◽  
Vol 37 (1) ◽  
pp. 116-122 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Dynamic Deformation ◽  
Three Dimensional ◽  
Elastic Solid ◽  
System Of Difference Equations ◽  
Order Accuracy ◽  
First Order ◽  
Dynamic Elasticity ◽  
Approximate Integral ◽  
Explicit Determination

The equations governing the dynamic deformation of an elastic solid are considered as a symmetric hyperbolic system of linear first-order partial-differential equations. The characteristic properties of the system are determined and a numerical method for obtaining the solution of mixed initial and boundary-value problems in elastodynamics is presented. The method, based on approximate integral relations along bicharacteristics, is an extension of the method proposed by Clifton for plane problems in dynamic elasticity and provides a system of difference equations, with second-order accuracy, for the explicit determination of the solution. Application of the method to a problem which has a known solution provides numerical evidence of the convergence and stability of the method.

scholarly journals Stability Regions of Fractional First Order Controllers Applied to Fractional Order Delay Systems

2021 ◽  
Vol 15 ◽  
pp. 86-90
Author(s):  
Karim Saadaoui
Keyword(s):  
Fractional Order ◽  
Real Root ◽  
Delay Systems ◽  
Time Delay Systems ◽  
Complex Root ◽  
Stability Regions ◽  
First Order ◽  
The Stability ◽  
Fractional Order Controller ◽  
Fractional Controller

This paper focuses on the problem of stabilizing fractional order time delay systems by fractional first order controllers. A solution is proposed to find the set of all stability regions in the controller’s parameter space. The D-decomposition method is employed to find the real root boundary and complex root boundaries which are used to identify the stability regions. Illustrative examples are given to show the effectiveness of the proposed approach, and it is remarked that the stability region obtained for the fractional order controller is larger than the non-fractional controller.

scholarly journals Optimum Techniques for the Conversion of Space Rectangular and Curvilinear Coordinates

2019 ◽  
Vol 4 (10) ◽  
pp. 147-151
Author(s):  
Gafar Suara ◽  
Timothy Oluwadare Idowu
Keyword(s):  
The Other ◽  
Curvilinear Coordinates ◽  
High Quality ◽  
Order Accuracy ◽  
First Order ◽  
Optimum Technique ◽  
Rectangular Coordinates ◽  
Optimum Model ◽  
Reduced Coordinates ◽  
Coordinate Conversion

Conversion between space rectangular (X, Y, Z) and curvilinear (φ, λ, h) coordinates is an important task in the field of Surveying, geodesy, positioning, navigation, mapping etc. Different techniques which include iterative methods, non-iterative techniques and closed form algebraic methods have been applied over the years to carry out the coordinate conversion. However, the results obtained using these techniques are deficient in one way or the other due to the inherent limitations such as inability to produce results for curvilinear coordinates when the values of X, Y and Z are subsequently or simultaneously equal to zero. Therefore, this study attempts to put forth an optimum coordinate conversion technique between space rectangular and curvilinear coordinates. The data used are coordinates of points which include the space rectangular coordinates and their equivalent curvilinear coordinates. They were observed and processed in Nigeria using Doppler 9 software by African Doppler Survey (ADOS) and they were confirmed to be of first order accuracy and hence of high quality. The data processing involved the design of the optimum techniques equations, coding of the algorithms and necessary computations to obtain results. Analyzing the results obtained, it can be inferred that the designed optimum model has successfully carried out the conversion between space rectangular and curvilinear coordinates. Therefore, the optimum technique model is recommended for use for the conversions from Space rectangular coordinates to Geocentric, Geodetic, Reduced coordinates and vice versa.

An Interpolation-Based Method for Multidimensional Extrapolation

2014 ◽  
Vol 1079-1080 ◽  
pp. 654-659
Author(s):  
Ru Chao Shi ◽  
Yong Chi Li
Keyword(s):  
Numerical Results ◽  
Numerical Implementation ◽  
Order Accuracy ◽  
First Order ◽  
Numerical Tests ◽  
Pde Method ◽  
Theoretical Results

This paper presents an interpolation-based method for multidimensional extrapolation. A series of interpolation formulations are proposed to extrapolate functions normal to the interface between two regions. Theoretical proofs and relevant analysis are also presented. The method developed maintains the characteristics of implicit interface. The interface inside every cell is treated smoothly by assuming that curvature is equal everywhere. The method developed and numerical results are verified by comparing to the results by PDE method and theoretical results. Numerical tests demonstrate that the method developed is first-order accuracy and also more efficient in numerical implementation and more accurate than PDE method.

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