Probabilistic derivation of the stability condition of Richardson’s explicit finite difference equation for the diffusion equation

1984 ◽  
Vol 52 (3) ◽  
pp. 267-267 ◽  
Author(s):  
Bernard Kaplan
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
Diffusion Equation ◽  
Stability Condition ◽  
Finite Difference Equation ◽  
Explicit Finite Difference ◽  
The Stability

Removing the stability limit of the time–space domain explicit finite difference schemes for acoustic modeling with stability condition-based spatial operators

Geophysics ◽  
2021 ◽  
pp. 1-82
Author(s):  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Stability Condition ◽  
Finite Difference Schemes ◽  
Grid Spacing ◽  
Time Step ◽  
Space Domain ◽  
Wavenumber Range ◽  
Time Space ◽  
Explicit Finite Difference ◽  
The Stability

The time step and grid spacing in explicit finite-difference (FD) modeling are constrained by the Courant-Friedrichs-Lewy (CFL) condition. Recently, it has been found that spatial FD coefficients may be designed through simultaneously minimizing the spatial dispersion error and maximizing the CFL number. This allows one to stably use a larger time step or a smaller grid spacing than usually possible. However, when using such a method, only second-order temporal accuracy is achieved. To address this issue, I propose a method to determine the spatial FD coefficients, which simultaneously satisfy the stability condition of the whole wavenumber range and the time–space domain dispersion relation of a given wavenumber range. Therefore, stable modeling can be performed with high-order spatial and temporal accuracy. The coefficients can adapt to the variation of velocity in heterogeneous models. Additionally, based on the hybrid absorbing boundary condition, I develop a strategy to stably and effectively suppress artificial reflections from the model boundaries for large CFL numbers. Stability analysis, accuracy analysis and numerical modeling demonstrate the accuracy and effectiveness of the proposed method.

The Alekseev–Mikhailenko method applied to P–SV wave propagation in an elastic medium

1988 ◽  
Vol 25 (2) ◽  
pp. 226-234
Author(s):  
L. J. Pascoe ◽  
F. Hron ◽  
P. F. Daley
Keyword(s):  
Wave Propagation ◽  
Finite Difference ◽  
Half Space ◽  
Elastic Half Space ◽  
Low Velocity ◽  
Seismic Modelling ◽  
Wave Propagation Problem ◽  
Modelling Techniques ◽  
Explicit Finite Difference ◽  
The Stability

The Alekseev–Mikhailenko method (AMM) is the name given to a series of algorithms that use one or more finite spatial transforms to reduce the dimensionality of a wave-propagation problem to that of one space dimension and time. This reduced equation is then solved using finite-difference techniques, and the space–time solution is recovered by applying inverse finite spatial transform(s). In this paper the elastodynamic wave equation that governs the coupled P–Sv motion in an isotropic, vertically inhomogeneous elastic half space is investigated using the AMM. Two types of impulsive body forces that may be used to excite the medium are examined, as is the problem of obtaining accurate transformed finite-difference analogues at the free surface. The second of these is accomplished by introducing the boundary conditions that the shear and normal stress must vanish here and by incorporating their transforms into the transformed elastodynamic equations. The stability criterion for the explicit finite-difference method is given cursory treatment, as detailed discussion of this aspect may be found in many texts that deal with the subject of finite differences.A coal-seam model (two thin, low-velocity layers embedded in a half space) illustrates the method. Both horizontal and vertical seismic traces are computed for this model and the results examined in relation to other seismic-modelling techniques.

Sur l'existence d'une longueur élémentaire et d'un intervalle de temps élémentaire

1978 ◽  
Vol 56 (8) ◽  
pp. 1109-1115 ◽  
Author(s):  
Robert Lacroix
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
One Dimension ◽  
Time Interval ◽  
Finite Difference Equation ◽  
Elementary Length

We have briefly examined several studies which have been made concerning the introduction of an elementary length l0 and an elementary time interval t0 into physical theories. We have discussed the arguments which we have found, arguments formulated by other authors, and which support the hypotheses concerning the existence of l0 and of t0. A finite difference equation is proposed and the solutions of some problems of movement in one dimension are given.

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