Elastic wave propagation using fully vectorized high order finite‐difference algorithms

Geophysics ◽  
1992 ◽  
Vol 57 (2) ◽  
pp. 218-232 ◽  
Author(s):  
A. Vafidis ◽  
F. Abramovici ◽  
E. R. Kanasewich
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Second Order ◽  
Finite Difference Schemes ◽  
Difference Operators ◽  
Two Dimensional ◽  
Elastic Wave Equation ◽  
The Matrix ◽  
Comparison Of The Results ◽  
High Order Finite Difference

Two finite‐difference schemes for solving the elastic wave equation in heterogeneous two‐dimensional media are implemented on a vector computer. A modified Lax‐Wendroff scheme that is second‐order accurate both in time and space and is a version of the MacCormack scheme that is second‐order accurate in time and fourth‐order in space. The algorithms are based on the matrix times vector by diagonals technique that is fully vectorized and is described using a novel notation for vector supercomputer operations. The technique described can be implemented on a vector processor of modest dimensions and increase the applicability of finite differences. The two difference operators are compared and the programs are tested for a simple case of standing sinusoidal waves for which the exact solution is known and also for a two‐layer model with a line source. A comparison of the results for an actual well‐to‐well experiment verifies the usefulness of the two‐dimensional approach in modeling the results.

scholarly journals Finite-Difference Simulation of Elastic Wave with Separation in Pure P- and S-Modes

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ke-Yang Chen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Computing Time ◽  
Staggered Grid ◽  
Difference Operators ◽  
Elastic Wave Equation ◽  
Absorbing Boundary ◽  
Wave Separation ◽  
Time Requirements

Elastic wave equation simulation offers a way to study the wave propagation when creating seismic data. We implement an equivalent dual elastic wave separation equation to simulate the velocity, pressure, divergence, and curl fields in pure P- and S-modes, and apply it in full elastic wave numerical simulation. We give the complete derivations of explicit high-order staggered-grid finite-difference operators, stability condition, dispersion relation, and perfectly matched layer (PML) absorbing boundary condition, and present the resulting discretized formulas for the proposed elastic wave equation. The final numerical results of pure P- and S-modes are completely separated. Storage and computing time requirements are strongly reduced compared to the previous works. Numerical testing is used further to demonstrate the performance of the presented method.

The pseudospectral method: Accurate representation of interfaces in elastic wave calculations

Geophysics ◽  
1988 ◽  
Vol 53 (5) ◽  
pp. 625-637 ◽  
Author(s):  
Bengt Fornberg
Keyword(s):  
Finite Difference ◽  
Elastic Wave ◽  
Finite Difference Methods ◽  
Pseudospectral Method ◽  
Mapping Method ◽  
Elastic Wave Equation ◽  
Finite Difference Approximations ◽  
Wave Trains ◽  
Discontinuous Medium ◽  
High Order Finite Difference

When finite‐difference methods are used to solve the elastic wave equation in a discontinuous medium, the error has two dominant components. Dispersive errors lead to artificial wave trains. Errors from interfaces lead to circular wavefronts emanating from each location where the interface appears “jagged” to the rectangular grid. The pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy. With this method, dispersive errors are essentially eliminated. The mappings introduced in this paper also eliminate the other dominant error source. Test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost. Although the mapping method is described here in connection with the pseudospectral method, it can also be used with high‐order finite‐difference approximations.

An implicit finite‐difference formulation of the elastic wave equation

Geophysics ◽  
1982 ◽  
Vol 47 (11) ◽  
pp. 1521-1526 ◽  
Author(s):  
Steven H. Emerman ◽  
W. Schmidt ◽  
R. A. Stephen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Large Time ◽  
Small Time ◽  
Finite Difference Schemes ◽  
Elastic Wave Equation ◽  
Stability Range ◽  
Implicit Finite Difference ◽  
The Stability

The problem with existing finite‐difference formulations of the elastic wave equation is that they have a limited stability range, and the necessity of taking small time steps can result in excessively high computation costs. It is possible to formulate a finite‐difference scheme which is stable for arbitrarily large time steps. However, the solutions obtained by the unconditionally stable scheme are unacceptably inaccurate for time steps outside the stability range of finite‐difference schemes currently in use.

scholarly journals A new fourth-order explicit group method in the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Muhammad Asim Khan ◽  
Norhashidah Hj. Mohd Ali ◽  
Nur Nadiah Abd Hamid
Keyword(s):  
Finite Difference ◽  
Stokes Problem ◽  
Stability And Convergence ◽  
Second Grade ◽  
Group Method ◽  
Two Dimensional ◽  
Second Grade Fluid ◽  
The Matrix ◽  
Grade Fluid ◽  
Generalized Second Grade Fluid

Abstract In this article, a new explicit group iterative scheme is developed for the solution of two-dimensional fractional Rayleigh–Stokes problem for a heated generalized second-grade fluid. The proposed scheme is based on the high-order compact Crank–Nicolson finite difference method. The resulting scheme consists of three-level finite difference approximations. The stability and convergence of the proposed method are studied using the matrix energy method. Finally, some numerical examples are provided to show the accuracy of the proposed method.

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