ACCURACY OF FINITE‐DIFFERENCE MODELING OF THE ACOUSTIC WAVE EQUATION

Geophysics ◽  
1974 ◽  
Vol 39 (6) ◽  
pp. 834-842 ◽  
Author(s):  
R. M. Alford ◽  
K. R. Kelly ◽  
D. M. Boore
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Order Accuracy ◽  
Order Scheme ◽  
The Finite Difference Method ◽  
Half Power ◽  
Grid Dispersion ◽  
Grid Interval

Recent interest in finite‐difference modeling of the wave equation has raised questions regarding the degree of match between finite‐difference solutions and solutions obtained by the more classical analytical approaches. This problem is studied by means of a comparison of seismograms computed for receivers located in the vicinity of a 90-degree wedge embedded in an infinite two‐dimensional acoustic medium. The calculations were carried out both by the finite‐difference method and by a more conventional eigenfunction expansion technique. The results indicate the solutions are in good agreement provided that the grid interval for the finite‐difference method is sufficiently small. If the grid is too coarse, the signals computed by the finite‐difference method become strongly dispersed, and agreement between the two methods rapidly deteriorates. This effect, known as “grid dispersion,” must be taken into account in order to avoid erroneous interpretation of seismograms obtained by finite‐difference techniques. Both second‐order accuracy and fourth‐order accuracy finite‐difference algorithms are considered. For the second‐order scheme, a good rule of thumb is that the ratio of the upper half‐power wavelength of the source to the grid interval should be of the order of ten or more. For the fourth‐order scheme, it is found that the grid can be twice as coarse (five or more grid points per upper half‐power wavelength) and good results are still obtained. Analytical predictions of the effect of grid dispersion are presented; these seem to be in agreement with the experimental results.

Higher Order Computations of the Convection-Diffusion Equation With Curved Boundaries

2003 ◽  
Author(s):  
Heejin Lee ◽  
Michael M. Chen
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
High Order ◽  
Complex Geometries ◽  
Third Order ◽  
Order Accuracy ◽  
Curved Boundaries ◽  
Structured Grid ◽  
The Finite Difference Method

In computational heat transfer and fluid mechanics, high order accuracy methods are desirable in order to reduce computational effort or to obtain more accurate solutions for a given mesh coarseness. On structured grids, the finite difference method is especially easy for deriving and implementing higher order schemes. In spite of this advantage, for complex geometries high order schemes have not been attractive due to the restriction of the structured grid in dealing with curved boundaries. Therefore, for complex geometries most computational methods are based on finite element or finite volume methods with unstructured or boundary-fitted mesh at the expense of difficult and complicated implementation. For this reason, few computations for complex geometries have attempted more than near-second-order accuracy. In our paper, we demonstrate a high order scheme to deal with curved boundaries of complex geometries in Cartesian coordinate system using the finite difference method, taking advantages of the ease and simplicity of structured grid. The method is based on an extension of the full second order methods presented previously by Jung et al. [2000] and Lee and Chen [2002]. The temperature distributions and maximum errors in a cylindrical solid and an annulus where the velocity distribution is given were calculated with a third order accurate scheme, and compared with exact solutions. Theoretical derivations and numerical experiments show that true third order accuracy have been attained in advection-diffusion problems with curved boundaries. The results reinforce the assertion that the same concepts can be extended to any order accuracy so far as such accuracy is deemed desirable for the problem of interest.

scholarly journals A new formulation of finite difference and finite volume methods for solving a space fractional convection–diffusion model with fewer error estimates

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reem Edwan ◽  
Shrideh Al-Omari ◽  
Mohammed Al-Smadi ◽  
Shaher Momani ◽  
Andreea Fulga
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Diffusion Equation ◽  
Finite Volume ◽  
Difference Method ◽  
Order Accuracy ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
The Finite Difference Method ◽  
Volume Method

AbstractConvection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional order using the finite volume method and the finite difference method. In this context, we present an alternative way for estimating the space fractional derivative by utilizing the fractional Grünwald formula. The proposed methods are conditionally stable with second-order accuracy in space and first-order accuracy in time. Many comparisons are performed to display reliability and capability of the proposed methods. Furthermore, several results and conclusions are provided to indicate appropriateness of the finite volume method in solving the space fractional convection–diffusion equation compared with the finite difference method.

APPLICATION OF A NUMERICAL-ANALYTICAL METHOD FOR SOLVING A ONE-DIMENSIONAL WAVE EQUATION

Akustika ◽  
2021 ◽  
pp. 22
Author(s):  
Vladimir Mondrus ◽  
Dmitrii Sizov
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Analytical Method ◽  
One Dimensional ◽  
Numerical Analytical Method ◽  
The Finite Difference Method ◽  
Dimensional Wave

The article contains a solution to the problem of wave propagation in a one-dimensional rod from the initial impact. A numerical-analytical method is used to solve the problem. The numerical part of the method is based on the application of the idea of the finite difference method. The analytical part uses the concept of Green’s function to solve the problem in terms of the spatial coordinate in the considered area. The results include graphs of the solution obtained at different points in time.

scholarly journals Splitting the One-Dimensional Wave Equation. Part I: Solving by Finite-Difference Method and Separation Variables

2020 ◽  
Vol 17 (2(SI)) ◽  
pp. 0675
Author(s):  
Shilan Othman Hussein
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Boundary Data ◽  
Zeroth Order ◽  
Output Force ◽  
Ill Posed ◽  
The Finite Difference Method ◽  
The One

In this study, an unknown force function dependent on the space in the wave equation is investigated. Numerically wave equation splitting in two parts, part one using the finite-difference method (FDM). Part two using separating variables method. This is the continuation and changing technique for solving inverse problem part in (1,2). Instead, the boundary element method (BEM) in (1,2), the finite-difference method (FDM) has applied. Boundary data are in the role of overdetermination data. The second part of the problem is inverse and ill-posed, since small errors in the extra boundary data cause errors in the force solution. Zeroth order of Tikhonov regularization, and several parameters of regularization are employed to decrease errors for output force solution. It is obvious from figures how error affects the results and zeroth order stables the solution.

scholarly journals WIDEBAND SATCOM MODEL: EVALUATION OF NUMERICAL ACCURACY AND EFFICIENCY

2020 ◽  
Vol V-3-2020 ◽  
pp. 105-110
Author(s):  
A. J. Knisely ◽  
A. J. Terzuoli
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Spectral Method ◽  
Difference Method ◽  
Signal Propagation ◽  
Input Function ◽  
Propagation Model ◽  
Phase Screen ◽  
The Finite Difference Method

Abstract. The spectral method is typically applied as a simple and efficient method to solve the parabolic wave equation in phase screen scintillation models. The critical factors that can greatly affect the spectral method accuracy is the uniformity and smoothness of the input function. This paper observes these effects on the accuracy of the finite difference and the spectral methods applied to a wideband SATCOM signal propagation model simulated in the ultra-high frequency (UHF) band. The finite difference method uses local pointwise approximations to calculate a derivative. The spectral method uses global trigonometric interpolants that achieve remarkable accuracy for continuously differentiable functions. The differences in accuracy are presented for a Gaussian lens and Kolmogorov phase screen. The results demonstrate loss of accuracy in each method when a phase screen is applied, despite the spectral method's computational efficiency over the finite difference method. These results provide meaningful insights when discretizing an interior domain and solving the parabolic wave equation to obtain amplitude and phase of a signal perturbation.

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