scholarly journals Approximation to Hadamard Derivative via the Finite Part Integral

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 983 ◽  
Author(s):  
Chuntao Yin ◽  
Changpin Li ◽  
Qinsheng Bi
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Numerical Methods ◽  
Three Dimensional ◽  
Cylindrical Wave ◽  
Physical Models ◽  
Finite Part ◽  
Numerical Examples ◽  
Crack Problems ◽  
Hadamard Derivative

In 1923, Hadamard encountered a class of integrals with strong singularities when using a particular Green’s function to solve the cylindrical wave equation. He ignored the infinite parts of such integrals after integrating by parts. Such an idea is very practical and useful in many physical models, e.g., the crack problems of both planar and three-dimensional elasticities. In this paper, we present the rectangular and trapezoidal formulas to approximate the Hadamard derivative by the idea of the finite part integral. Then, we apply the proposed numerical methods to the differential equation with the Hadamard derivative. Finally, several numerical examples are displayed to show the effectiveness of the basic idea and technique.

scholarly journals A Comparison of Splitting Techniques for 3D Complex Padé Fourier Finite Difference Migration

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Jessé C. Costa ◽  
Débora Mondini ◽  
Jörg Schleicher ◽  
Amélia Novais
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Computational Cost ◽  
Three Dimensional ◽  
Depth Level ◽  
Numerical Examples ◽  
The Cost ◽  
Comparable Quality ◽  
3D Problem ◽  
Dimensional Wave

Three-dimensional wave-equation migration techniques are still quite expensive because of the huge matrices that need to be inverted. Several techniques have been proposed to reduce this cost by splitting the full 3D problem into a sequence of 2D problems. We compare the performance of splitting techniques for stable 3D Fourier finite-difference (FFD) migration techniques in terms of image quality and computational cost. The FFD methods are complex Padé FFD and FFD plus interpolation, and the compared splitting techniques are two- and four-way splitting as well as alternating four-way splitting, that is, splitting into the coordinate directions at one depth and the diagonal directions at the next depth level. From numerical examples in homogeneous and inhomogeneous media, we conclude that, though theoretically less accurate, alternate four-way splitting yields results of comparable quality as full four-way splitting at the cost of two-way splitting.

scholarly journals Boundary conditions for the wave equation

1933 ◽  
Vol 141 (843) ◽  
pp. 216-217
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
Existence Theorems ◽  
Theoretical Discussion ◽  
Wave Mechanics ◽  
Eigen Values ◽  
Eigen Solutions ◽  
Good Agreement

A single electron in the field of two fixed nuclei, constituting the idealized hydrogen molecular ion, provides the simplest case for the application of wave mechanics to molecular, as distinct from atomic, problems. The most extensive theoretical discussion of the corresponding wave equation has been given by A. H. Wilson in these 'Proceedings.’ He was led to conclude that this equation possesses no eigen-solutions satisfying the usual boundary conditions for an atomic problem. Subsequent investigators have succeeded, however, in obtaining by numerical methods eigen-values in good agreement with observed values of the energy. But, with the exception of Teller, they appear not to have taken account of Wilson’s result. It is therefore worth while to investigate the existence of their solutions and to clear up, if possible, any doubt as to the applicability of the familiar boundary conditions to this type of problem. The usual existence theorems for eigen-values apply only to boundary conditions at ordinary points of the differential equation. The difficulty in cases like Wilson’s equation is that the conditions are given at singular points.

scholarly journals Four free surface algorithms for the 3D Navier–Stokes equations

2015 ◽  
Vol 17 (6) ◽  
pp. 845-856 ◽  
Author(s):  
Nils Reidar B. Olsen
Keyword(s):  
Wave Equation ◽  
Numerical Methods ◽  
Water Surface ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Free Water ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Free Water Surface

Four algorithms are described for computing a steady free water surface with the solution of the three-dimensional (3D) Navier–Stokes equations. The numerical methods are used in hydraulic engineering cases, typically spillways and river modelling. The algorithms were tested against a laboratory experiment of a v-shaped broad-crested weir. The complex geometry of the weir introduced three-dimensional effects, which the numerical methods handled with varying degrees of success. One of the methods tested was the classical volume of fluid (VOF) approach, implemented in the OpenFOAM software with a fixed grid. The other three algorithms used an adaptive grid that followed the free water surface. These methods were coded in the SSIIM 2 program and were based on water continuity, pressure differences and an implicit solution of the diffusive wave equation. The VOF method gave the best results compared with the experiments. However, this method requires a very short time step. Two of the investigated methods compute the water surface location implicitly and can therefore use a much longer time step. The method based on the diffusive wave equation has the disadvantage that the results depend on a calibrated friction factor. All four methods predicted the water depth over the weir with an average accuracy below 14%.

scholarly journals Three-Dimensional Complex Padé FD Migration: Splitting and Corrections

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
D. Mondini ◽  
J. C. Costa ◽  
J. Schleicher ◽  
A. Novais
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Computational Cost ◽  
Three Dimensional ◽  
Cost Effective ◽  
Effective Strategy ◽  
Depth Level ◽  
Numerical Examples ◽  
3D Problem ◽  
Dimensional Wave

Three-dimensional wave-equation migration techniques are still quite expensive because of the huge matrices that need to be inverted. Several techniques have been proposed to reduce this cost by splitting the full 3D problem into a sequence of 2D problems. To reduce errors, the Li correction is applied at regular multiples of depth extrapolation increment. We compare the performance of splitting techniques in wave propagation for 3D finite-difference (FD) migration in terms of image quality and computational cost. We study the behaviour of the complex Padé approximation in combination with two- and alternating four-way splitting, that is, splitting into the coordinate directions at one depth and the diagonal directions at the next depth level. We also extend the Li correction for use with the complex Padé expansion and diagonal directions. From numerical examples in inhomogeneous media, we conclude that alternate four-way splitting is the most cost-effective strategy to reduce numerical anisotropy in complex Padé 3D FD migration.

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