scholarly journals Viscoelastic finite‐difference modeling

Geophysics ◽  
1994 ◽  
Vol 59 (9) ◽  
pp. 1444-1456 ◽  
Author(s):  
Johan O. A. Robertsson ◽  
Joakim O. Blanch ◽  
William W. Symes
Keyword(s):  
Finite Difference ◽  
Good Reason ◽  
Viscoelastic Model ◽  
Finite Difference Schemes ◽  
Order Accuracy ◽  
Absorbing Boundaries ◽  
Mechanical Waves ◽  
Staggered Scheme ◽  
Anelastic Behavior ◽  
Grid Points

Real earth media disperse and attenuate propagating mechanical waves. This anelastic behavior can be described well by a viscoelastic model. We have developed a finite‐difference simulator to model wave propagation in viscoelastic media. The finite‐difference method was chosen in favor of other methods for several reasons. Finite‐difference codes are more portable than, for example, pseudospectral codes. Moreover, finite‐difference schemes provide a convenient environment in which to define complicated boundaries. A staggered scheme of second‐order accuracy in time and fourth‐order accuracy in space appears to be optimally efficient. Because of intrinsic dispersion, no fixed grid points per wavelength rule can be given; instead, we present tables, which enable a choice of grid parameters for a given level of accuracy. Since the scheme models energy absorption, natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary. The viscoelastic scheme is only marginally more expensive than analogous elastic schemes. The efficient implementation of absorbing boundaries may therefore be a good reason for also using the viscoelastic scheme in purely elastic simulations. We illustrate our method and the importance of accurately modeling anelastic media through 2-D and 3-D examples from shallow marine environments.

scholarly journals Third-Order Finite-Difference Schemes on Icosahedral-Type Grids on the Sphere

2008 ◽  
Vol 136 (7) ◽  
pp. 2683-2698 ◽  
Author(s):  
J. Steppeler ◽  
P. Rípodas ◽  
B. Jonkheid ◽  
S. Thomas
Keyword(s):  
Finite Difference ◽  
Fourth Order ◽  
Practical Method ◽  
Two Dimensions ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Limited Area ◽  
Third Order ◽  
Grid Points ◽  
Time Accuracy

Abstract A practical method is proposed to achieve high-order finite-difference schemes on grids that are quasi-homogeneous on the sphere. A family of grids is used that are characterized by the parameter NP, which can take on values of 3, 4, and 5, etc. The parameter NP is the number of grid patches meeting at the Poles. For NP = 3 the cube sphere grid is obtained and for NP = 5 the icosahedron is obtained. While the grid construction method is valid for all values of NP, the tests performed in this paper concern only the case NP = 5 (i.e., the icosahedron). For each of the rhomboidal patches, the grid is created by connecting points on opposing sides of the rhomboid by great circles. This offers the possibility to obtain derivatives for a line of grid points along a great circle in the classical way. Therefore, it becomes possible to use well-known spatial discretizations from limited-area models. Local models can be transferred to the sphere with rather limited effort. The method was tested using the fourth-order Runge–Kutta integration method and fourth-order spatial differencing. At patch limits, boundary values are obtained using third-order serendipity interpolation, giving the scheme an overall space–time accuracy of 3. The serendipity interpolation is quite efficient. Third-order interpolation in two dimensions is achieved by a set of linear interpolations and a number of function evaluations. All coefficients can be precomputed. The third-order convergence is demonstrated by numerical experiments using Williamson’s test cases 2 and 6.

Modeling elastic fields across irregular boundaries

Geophysics ◽  
1992 ◽  
Vol 57 (9) ◽  
pp. 1189-1193 ◽  
Author(s):  
Francis Muir ◽  
Joe Dellinger ◽  
John Etgen ◽  
Dave Nichols
Keyword(s):  
Finite Difference ◽  
Computer Modeling ◽  
Elastic Constants ◽  
Finite Difference Schemes ◽  
Geological Model ◽  
Rectangular Grid ◽  
Elastic Fields ◽  
The Earth ◽  
Difference Grid ◽  
Grid Points

Geologists often see the earth as homogeneous blocks separated by smoothly curving boundaries. In contrast, computer modeling algorithms based on finite‐difference schemes require elastic constants to be specified on the vertices of a regular rectangular grid. How can we convert a continuous geological model into a form suitable for a finite‐difference grid? One common way is to lay the finite‐difference grid down on the continuous geological model and use whatever elastic constants happen to lie beneath each of the grid points.

scholarly journals Effects of Physical Coordinate Clustering on Boundary Vorticity Approximations in von Mises Coordinates

2021 ◽  
Vol 16 ◽  
pp. 201-213
Author(s):  
M. H. Hamdan
Keyword(s):  
Finite Difference ◽  
Solid Boundary ◽  
Computational Domain ◽  
Physical Domain ◽  
Order Accuracy ◽  
First Order ◽  
Grid Points ◽  
Von Mises

Forward finite difference expressions of first-order accuracy for boundary vorticity on a solid boundary are evaluated in this work when the physical coordinates are clustered and mapped using von Mises coordinates. Results show that schemes using in-field grid points do not improve solutions obtained. Results also show that the finer the grid used in the physical domain, and the more clustered it is, improves the boundary vorticity values in the computational domain. The “best” expressions forward finite difference expressions are identified when two, three, four and five grid points are used.

Finite difference scheme for the time-space fractional diffusion equations

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Jianxiong Cao ◽  
Changpin Li
Keyword(s):  
Finite Difference ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Stability And Convergence ◽  
Finite Difference Schemes ◽  
Order Accuracy ◽  
Fractional Diffusion Equations ◽  
Time Space ◽  
Liouville Fractional Derivative ◽  
The Stability

AbstractIn this paper, we derive two novel finite difference schemes for two types of time-space fractional diffusion equations by adopting weighted and shifted Grünwald operator, which is used to approximate the Riemann-Liouville fractional derivative to the second order accuracy. The stability and convergence of the schemes are analyzed via mathematical induction. Moreover, the illustrative numerical examples are carried out to verify the accuracy and effectiveness of the schemes.

On the internal interfaces in finite-difference schemes

Geophysics ◽  
2017 ◽  
Vol 82 (4) ◽  
pp. T159-T182 ◽  
Author(s):  
Rune Mittet
Keyword(s):  
Finite Difference ◽  
Wave Equations ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Equal Sign ◽  
Fine Grid ◽  
Elastic Wave Equations ◽  
Low Pass ◽  
Grid Points ◽  
Locally Homogeneous

Implementing sharp internal interfaces in finite-difference schemes with high spatial accuracy is challenging. The propagation of fields in a locally homogeneous part of a model can be performed with spectral accuracy. The implementations of interfaces are generally considered accurate to, at best, second order. This situation can be improved by a proper band limitation of the simulation grid. Interfaces can be located anywhere on the grid; however, the fine detail information regarding the interface location must be imprinted correctly in the coarse simulation grid. This can be done by starting out with a representation of a sharp material jump in the wavenumber domain and limiting the highest wavenumber to the maximum wavenumber allowed for the simulation grid. The resulting wavenumber representation is then transformed to the space domain. An alternative procedure is to create a fine grid model that is low-pass filtered to remove wavenumbers above the maximum wavenumber allowed for the coarse simulation grid. The fine grid is thereafter sampled at the required coordinates for the coarse simulation grid. An accurate and flexible interface implementation is a requisite for reducing staircase diffractions in higher dimensional finite-difference simulations. Our strategy achieves this. The frequency content of the source must be constrained to a level in which the spatial sampling is at approximately four to five grid points per shortest wavelength. Simulation results indicate that the implementation of the interface is accurate to at least the sixth order for large contrasts. Our method can be used for all systems of partial differential equations that formally can be expressed as a material parameter times a dynamic field on one side of the equal sign and with spatial derivatives on the other side of the equal sign. For geophysical simulations, the most important cases will be the Maxwell equations and the acoustic and elastic wave equations.

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