A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. A1-A6 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Numerical Modeling ◽  
Wave Equation ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Computational Domain ◽  
Perfect Absorption ◽  
Transition Area ◽  
Efficient Scheme ◽  
Grid Points ◽  
The One

We propose an efficient scheme to absorb reflections from the model boundaries in numerical solutions of wave equations. This scheme divides the computational domain into boundary, transition, and inner areas. The wavefields within the inner and boundary areas are computed by the wave equation and the one-way wave equation, respectively. The wavefields within the transition area are determined by a weighted combination of the wavefields computed by the wave equation and the one-way wave equation to obtain a smooth variation from the inner area to the boundary via the transition zone. The results from our finite-difference numerical modeling tests of the 2D acoustic wave equation show that the absorption enforced by this scheme gradually increases with increasing width of the transition area. We obtain equally good performance using pseudospectral and finite-element modeling with the same scheme. Our numerical experiments demonstrate that use of 10 grid points for absorbing edge reflections attains nearly perfect absorption.

The Dispersion Electrons in the One-Electron Problem

1929 ◽  
Vol 25 (3) ◽  
pp. 323-330
Author(s):  
J. Hargreaves
Keyword(s):  
Quantum Mechanics ◽  
Wave Equation ◽  
Dipole Moment ◽  
Optical Spectrum ◽  
Wave Equations ◽  
Simple Wave ◽  
Incoherent Scattering ◽  
Central Field ◽  
Dispersion Formula ◽  
The One

In this paper we use Dirac's relativity quantum mechanics to derive the well-known Kramers-Heisenberg dispersion formula for an atom with one electron. The treatment is not limited to the case of a central field, but is quite general. An expression is also obtained for the dipole moment to which is due the incoherent scattering. The formulae obtained are similar to those obtained by O. Klein for the case of a central field. We find in this way an explicit expression for f, the number of dispersion electrons for any line of the optical spectrum (being a measure of the intensity of the line), in terms of the solutions of the four wave equations of Dirac's theory. It is further shown that the sum of the number of dispersion electrons for any state of the atom is not exactly unity, but differs from it by an amount of the order of 10−4. The result Σf = 1 has been shown by London to hold exactly for the simple wave equation as originally given by Schrödinger. It is here shown that the exact relativity treatment has a very small effect.

Numerical simulation of seismic wave propagation in viscoelastic-anisotropic media using frequency-independent Q wave equation

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. WA1-WA10 ◽  
Author(s):  
Tieyuan Zhu
Keyword(s):  
Numerical Modeling ◽  
Wave Propagation ◽  
Wave Equation ◽  
Inhomogeneous Medium ◽  
Seismic Anisotropy ◽  
Numerical Solutions ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
Theoretical Formula ◽  
Frequency Independent

Seismic anisotropy is the fundamental phenomenon of wave propagation in the earth’s interior. Numerical modeling of wave behavior is critical for exploration and global seismology studies. The full elastic (anisotropy) wave equation is often used to model the complexity of velocity anisotropy, but it ignores attenuation anisotropy. I have presented a time-domain displacement-stress formulation of the anisotropic-viscoelastic wave equation, which holds for arbitrarily anisotropic velocity and attenuation [Formula: see text]. The frequency-independent [Formula: see text] model is considered in the seismic frequency band; thus, anisotropic attenuation is mathematically expressed by way of fractional time derivatives, which are solved using the truncated Grünwald-Letnikov approximation. I evaluate the accuracy of numerical solutions in a homogeneous transversely isotropic (TI) medium by comparing with theoretical [Formula: see text] and [Formula: see text] values calculated from the Christoffel equation. Numerical modeling results show that the anisotropic attenuation is angle dependent and significantly different from the isotropic attenuation. In synthetic examples, I have proved its generality and feasibility by modeling wave propagation in a 2D TI inhomogeneous medium and a 3D orthorhombic inhomogeneous medium.

scholarly journals Splitting the one-Dimensional Wave Equation, Part II: Additional Data are Given by an End Displacement Measurement

2021 ◽  
pp. 233-239
Author(s):  
Shilan Othman Hussein ◽  
Mohammed Sabah Hussein
Keyword(s):  
Wave Equation ◽  
Additional Data ◽  
Numerical Solutions ◽  
Separation Of Variables ◽  
Displacement Measurement ◽  
Tikhonov Regularization Method ◽  
Ill Posed ◽  
The One ◽  
Inverse Force ◽  
Force Problem

     In this research, an unknown space-dependent force function in the wave equation is studied. This is a natural continuation of [1] and chapter 2 of [2] and [3], where the finite difference method (FDM)/boundary element method (BEM), with the separation of variables method, were considered. Additional data are given by the one end displacement measurement. Moreover, it is a continuation of [3], with exchanging the boundary condition, where  are extra data, by the initial condition. This is an ill-posed inverse force problem for linear hyperbolic equation. Therefore, in order to stabilize the solution, a zeroth-order Tikhonov regularization method is provided. To assess the accuracy, the minimum error between exact and numerical solutions for the force is computed for various regularization parameters. Numerical results are presented and a good agreement was obtained for the exact and noisy data.

scholarly journals Modelling of Flood Wave Propagation with Wet-dry Front by One-dimensional Diffusive Wave Equation

2014 ◽  
Vol 61 (3-4) ◽  
pp. 111-125 ◽  
Author(s):  
Dariusz Gąsiorowski
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Numerical Solutions ◽  
Numerical Algorithms ◽  
Finite Volume Methods ◽  
Flood Wave ◽  
One Dimensional ◽  
Benchmark Solution ◽  
The One ◽  
Water Depths

Abstract A full dynamic model in the form of the shallow water equations (SWE) is often useful for reproducing the unsteady flow in open channels, as well as over a floodplain. However, most of the numerical algorithms applied to the solution of the SWE fail when flood wave propagation over an initially dry area is simulated. The main problems are related to the very small or negative values of water depths occurring in the vicinity of a moving wet-dry front, which lead to instability in numerical solutions. To overcome these difficulties, a simplified model in the form of a non-linear diffusive wave equation (DWE) can be used. The diffusive wave approach requires numerical algorithms that are much simpler, and consequently, the computational process is more effective than in the case of the SWE. In this paper, the numerical solution of the one-dimensional DWE based on the modified finite element method is verified in terms of accuracy. The resulting solutions of the DWE are compared with the corresponding benchmark solution of the one-dimensional SWE obtained by means of the finite volume methods. The results of numerical experiments show that the algorithm applied is capable of reproducing the reference solution with satisfactory accuracy even for a rapidly varied wave over a dry bottom.

Determination of finite difference coefficients for the acoustic wave equation using regularized least-squares inversion

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Yanfei Wang ◽  
Wenquan Liang ◽  
Zuhair Nashed ◽  
Changchun Yang
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Least Squares ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Spatial Domain ◽  
Variable Length ◽  
Staggered Grid ◽  
Regular Grid ◽  
Low Velocity

AbstractFinite difference (FD) solutions of wave equations have been proven useful in exploration seismology. To yield reliable and interpretable results, the numerically induced error should be minimized over a range of frequencies and angles of propagation. Grid dispersion is one of the key numerical problems and there exist some methods to solve this problem in the literature. Traditionally, the spatial FD operator coefficients are only determined in the spatial domain; however, the wave equation is solved in the temporal and spatial domain simultaneously. Recently, some methods based on the joint temporal-spatial domains have been proposed to address this problem. Variable length coefficients methods are proposed in the literature to improve efficiency while preserving accuracy by using longer operators in the low velocity regions and shorter operators in the high velocity regions. To cope with the ill-conditioning of the linear system induced by long stencil FD operators, we study in this paper a regularizing simplified least-squares model to minimize the phase velocity error in the joint temporal-spatial domain with a variable length of coefficients. Different from our previous study, we determine FD coefficients on the regular grid instead of on the staggered grid. Though the regular grid FD methods are less precise, however, with a little increase of the operator length, the precision can be improved. Stability of the numerical solutions is enhanced by the regularization. Numerical simulations made on one-dimensional to three-dimensional examples show that our scheme needs shorter operators and preserves accuracy compared with the previous methods.

Acoustic VTI modeling with a time-space domain dispersion-relation-based finite-difference scheme

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. A11-A17 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Numerical Modeling ◽  
Dispersion Relation ◽  
Finite Difference ◽  
Wave Equations ◽  
Computational Cost ◽  
Transversely Isotropic ◽  
Dispersion Analysis ◽  
Space Domain ◽  
Time Space ◽  
Grid Points

The finite-difference (FD) method is widely used in numerical modeling of wave equations. Conventional FD stencils for space derivatives are usually designed in the space domain. When they are used to solve wave equations, it is difficult to satisfy the dispersion relations exactly. We have designed a spatial FD stencil based on a time-space domain dispersion relation to simulate wave propagation in an acoustic vertically transversely isotropic (VTI) medium. Two-dimensional dispersion analysis and numerical modeling demonstrate that this stencil has greater precision than one used in a conventional FD, when the same number of grid points is used in the calculation of the spatial derivatives. Thus, the spatial FD stencil based on time-space domain dispersion relation can be used to replace a conventional one such that we can achieve greater accuracy with almost no increase in computational cost.

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Yuri Luchko ◽  
Francesco Mainardi
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Wave Equations ◽  
Propagation Speed ◽  
Fractional Diffusion ◽  
Maximum Point ◽  
Diffusion Wave ◽  
Constant Coefficients ◽  
The One ◽  
Signalling Problem

AbstractIn this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.

VOLUME SEGMENTATION BY POST-PROCESSING DATA FROM SIMULATION OF SOLIDIFICATION IN THE METAL CASTING PROCESS

2013 ◽  
Vol 04 (supp01) ◽  
pp. 1341005 ◽  
Author(s):  
ELIZABETH JACOB ◽  
M. MANOJ ◽  
ROSCHEN SASIKUMAR
Keyword(s):  
Spatial Data ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Solidification Process ◽  
Casting Process ◽  
Solidification Time ◽  
Computational Domain ◽  
Post Processing ◽  
Grid Points ◽  
Processing Techniques

In the process of interpreting simulation results, new post-processing techniques are developed. This work presents a post-processing method that analyzes the solidification pattern formed by simulation of the solidification process of molten metal in a mold to produce shaped castings. Simulations generally involve numerical solutions of differential equations which are discretized by dividing the three-dimensional computational domain into small finite volume elements using a 3D grid. The locations of the grid points and values of the solidification time at these locations are used to divide the spatial data into 3D sections such that starting from a hotspot location within the section that has high solidification time, there is a gradient outwards with lower values of solidification time. Each section is assumed to be fed by one or more feeders that must freeze only after the section has solidified completely. The volume of a feeder can be determined from the volume of the section it is supposed to feed. The volume and surface area of sections are determined approximately to calculate feeder size and dimensions. The post-processing algorithm is a simulation-based quantitative approach to feeder design which in conventional foundry practice has been more of an art than science. It is also general enough for use in other 3D segmentation applications.

A Bremmer series for radial wave equations

1977 ◽  
Vol 55 (24) ◽  
pp. 2150-2157 ◽  
Author(s):  
W. E. Couch ◽  
R. J. Torrence
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Wkb Approximation ◽  
Spherically Symmetric ◽  
Radial Wave ◽  
One Dimensional ◽  
Similar Series ◽  
Leading Term ◽  
The One ◽  
Radial Wave Equation

The Bremmer series solution of the one-dimensional Helmholtz equation with variable velocity is generalized to obtain a similar series for the radial wave equation with a spherically symmetric velocity function. Since the leading term of Bremmer's series is the one-dimensional WKB approximation, we obtain an approximation for the radial wave equation analogous to the WKB approximation.

On Numerical Solutions to the One-Dimensional Wave Equation

1954 ◽  
Vol 61 (2) ◽  
pp. 115 ◽  
Author(s):  
W. P. Reid
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave
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