Finite‐difference solution of the eikonal equation along expanding wavefronts

Geophysics ◽  
1992 ◽  
Vol 57 (3) ◽  
pp. 478-487 ◽  
Author(s):  
Fuhao Qin ◽  
Yi Luo ◽  
Kim B. Olsen ◽  
Wenying Cai ◽  
Gerard T. Schuster
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Scheme ◽  
Eikonal Equation ◽  
Computational Cost ◽  
Velocity Models ◽  
Large Velocity ◽  
Programming Effort ◽  
Finite Difference Solution ◽  
Solution Region

We show that a scheme to solve the 2-D eikonal equation by a finite‐difference method can violate causality for moderate to large velocity contrasts [Formula: see text]. As an alternative, we present a finite‐difference scheme in which the solution region progresses outward from an “expanding wavefront” rather than an “expanding square,” and therefore honors causality. Our method appears to be stable and reasonably accurate for a variety of velocity models with moderate to large velocity contrasts. The penalty is a large increase in computational cost and programming effort.

3-D traveltime computation using second‐order ENO scheme

Geophysics ◽  
1999 ◽  
Vol 64 (6) ◽  
pp. 1867-1876 ◽  
Author(s):  
Seongjai Kim ◽  
Richard Cook
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Eikonal Equation ◽  
Second Order ◽  
Order Accuracy ◽  
Unconditionally Stable ◽  
Velocity Models ◽  
Box Scheme ◽  
Initialization Scheme ◽  
Eno Scheme

We consider a second‐order finite difference scheme to solve the eikonal equation. Upwind differences are requisite to sharply resolve discontinuities in the traveltime derivatives, whereas centered differences improve the accuracy of the computed traveltime. A second‐order upwind essentially non‐oscillatory (ENO) scheme satisfies these requirements. It is implemented with a dynamic down ’n’ out (DNO) marching, an expanding box approach. To overcome the instability of such an expanding box scheme, the algorithm incorporates an efficient post sweeping (PS), a correction‐by‐iteration method. Near the source, an efficient and accurate mesh‐refinement initialization scheme is suggested for the DNO marching. The resulting algorithm, ENO-DNO-PS, turns out to be unconditionally stable, of second‐order accuracy, and efficient; for various synthetic and real velocity models having large contrasts, two PS iterations produce traveltimes accurate enough to complete the computation.

Finite‐difference solution of the eikonal equation using an efficient, first‐arrival, wavefront tracking scheme

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 632-643 ◽  
Author(s):  
Shunhua Cao ◽  
Stewart Greenhalgh
Keyword(s):  
Finite Difference ◽  
Eikonal Equation ◽  
Discretization Scheme ◽  
Efficiency Improvement ◽  
Average Value ◽  
Velocity Models ◽  
First Arrival ◽  
Finite Difference Solution ◽  
Corner Node ◽  
Difference Solution

First‐break traveltimes can be accurately computed by the finite‐difference solution of the eikonal equation using a new corner‐node discretization scheme. It offers accuracy advantages over the traditional cell‐centered node scheme. A substantial efficiency improvement is achieved by the incorporation of a wavefront tracking algorithm based on the construction of a minimum traveltime tree. For the traditional discretization scheme, an accurate average value for the local squared slowness is found to be crucial in stabilizing the numerical scheme for models with large slowness contrasts. An improved method based on the traditional discretization scheme can be used to calculate traveltimes in arbitrarily varying velocity models, but the method based on the corner‐node discretization scheme provides a much better solution.

Solution of the Eikonal equation by a finite‐difference method

1990 ◽  
Author(s):  
Fuhao Qin ◽  
Kim Bak Olsen ◽  
Yi Luo ◽  
Gerard T. Schuster
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Eikonal Equation

Upwind finite‐difference calculation of traveltimes

Geophysics ◽  
1991 ◽  
Vol 56 (6) ◽  
pp. 812-821 ◽  
Author(s):  
J. van Trier ◽  
W. W. Symes
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Conservation Law ◽  
Finite Difference Scheme ◽  
Eikonal Equation ◽  
Upwind Schemes ◽  
Similar Scheme ◽  
Difference Calculation ◽  
Flow Variables ◽  
Upwind Finite Difference

Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite‐difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first‐arrival‐time field. The upwind finite‐difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first‐order upwind finite‐difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of fluid flow by using only information taken from upstream in the fluid. Other common difference schemes are unstable, or overly dissipative, at shocks (discontinuities in flow variables), which are time gradient discontinuities in our approach to solving the eikonal equation.

Method of corrections by higher order differences for elliptic equations with variable coefficients

2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Givi Berikelashvili ◽  
Bidzina Midodashvili
Keyword(s):  
Dirichlet Problem ◽  
Elliptic Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Convergence Rate ◽  
Difference Scheme ◽  
Elliptic Equations ◽  
Variable Coefficients ◽  
Order Accuracy ◽  
Corrected Scheme

AbstractWe consider the Dirichlet problem for an elliptic equation with variable coefficients, the solution of which is obtained by means of a finite-difference scheme of second order accuracy. We establish a two-stage finite-difference method for the posed problem and obtain an estimate of the convergence rate consistent with the smoothness of the solution. It is proved that the solution of the corrected scheme converges at rate

scholarly journals CMMSE: A technique for generating adapted discretizations to solve partial differential equations with the generalized finite difference method

2021 ◽  
Author(s):  
Augusto César Ferreira ◽  
Miguel Ureña ◽  
HIGINIO RAMOS
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Meshless Method ◽  
Computational Cost ◽  
Partial Differential

The generalized finite difference method is a meshless method for solving partial differential equations that allows arbitrary discretizations of points. Typically, the discretizations have the same density of points in the domain. We propose a technique to get adapted discretizations for the solution of partial differential equations. This strategy allows using a smaller number of points and a lower computational cost to achieve the same accuracy that would be obtained with a regular discretization.

scholarly journals Finite Difference Solution to a Nonlinear Time-Fractional Logistic Model

2021 ◽  
Vol 13 (2) ◽  
pp. 60
Author(s):  
Yuanyuan Yang ◽  
Gongsheng Li
Keyword(s):  
Finite Difference ◽  
Theoretical Analysis ◽  
Difference Scheme ◽  
Logistic Model ◽  
Stability And Convergence ◽  
Numerical Examples ◽  
Implicit Finite Difference Scheme ◽  
Finite Difference Solution ◽  
Implicit Finite Difference ◽  
Convergence Of The Scheme

We set forth a time-fractional logistic model and give an implicit finite difference scheme for solving of the model. The L^2 stability and convergence of the scheme are proved with the aids of discrete Gronwall inequality, and numerical examples are presented to support the theoretical analysis.

scholarly journals Finite difference solution of plate bending using Wolfram Mathematica

2019 ◽  
Vol 13 (3) ◽  
pp. 241-247
Author(s):  
Katarina Pisačić ◽  
Marko Horvat ◽  
Zlatko Botak
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Rectangular Plate ◽  
Difference Method ◽  
Mapping Function ◽  
Equation System ◽  
Plate Bending ◽  
Wolfram Mathematica ◽  
Linear Equation System ◽  
Finite Difference Solution

This article describes the procedure of calculating deflection of rectangular plate using a finite difference method, programmed in Wolfram Mathematica. Homogenous rectangular plate under uniform pressure is simulated for this paper. In the introduction, basic assumptions are given and the problem is defined. Chapters that follow describe basic definitions for plate bending, deflection, slope and curvature. The following boundary condition is used in this article: rectangular plate is wedged on one side and simply supported on three sides. Using finite difference method, linear equation system is given and solved in Wolfram Mathematica. System of equations is built using the mapping function and solved with solve function. Solutions are given in the graphs. Such obtained solutions are compared to the finite element method solver NastranInCad.

An Algorithm for the Inverse Solution of Geodesic Sailing without Auxiliary Sphere

2014 ◽  
Vol 67 (5) ◽  
pp. 825-844 ◽  
Author(s):  
Wei-Kuo Tseng
Keyword(s):  
Inverse Problem ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Computer Algebra ◽  
Computer Algebra System ◽  
Computational Cost ◽  
Inverse Solution ◽  
Series Expansions ◽  
Trigonometric Functions ◽  
Innovative Method

An innovative algorithm to determine the inverse solution of a geodesic with the vertex or Clairaut constant located between two points on a spheroid is presented. This solution to the inverse problem will be useful for solving problems in navigation as well as geodesy. The algorithm to be described derives from a series expansion that replaces integrals for distance and longitude, while avoiding reliance on trigonometric functions. In addition, these series expansions are economical in terms of computational cost. For end points located at each side of a vertex, certain numerical difficulties arise. A finite difference method together with an innovative method of iteration that approximates Newton's method is presented which overcomes these shortcomings encountered for nearly antipodal regions. The method provided here, which does not involve an auxiliary sphere, was aided by the Computer Algebra System (CAS) that can yield arbitrarily truncated series suitable to the users accuracy objectives and which are limited only by machine precisions.

The role and application of entropy terms for acoustic wave modeling by the finite‐difference method

Geophysics ◽  
1984 ◽  
Vol 49 (9) ◽  
pp. 1457-1465 ◽  
Author(s):  
M. A. Dablain
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Acoustic Wave Propagation ◽  
Central Difference ◽  
Central Difference Scheme ◽  
One Dimensional ◽  
The Finite Difference Method

The significance of entropy‐like terms is examined within the context of the finite‐difference modeling of acoustic wave propagation. The numerical implications of dissipative mechanisms are tested for performance within two very distinct differencing algorithms. The two schemes which are reviewed with and without dissipation are (1) the standard central‐difference scheme, and (2) the Lax‐Wendroff two‐step scheme. Numerical results are presented comparing the short‐wavelength response of these schemes. In order to achieve this response, the linearized version of an exploding one‐dimensional source is used.

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