scholarly journals A finite difference method on layer-adapted meshes for an elliptic reaction-diffusion system in two dimensions

2008 ◽  
Vol 77 (264) ◽  
pp. 2085-2096 ◽  
Author(s):  
R. Bruce Kellogg ◽  
Torsten Linss ◽  
Martin Stynes
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Reaction Diffusion ◽  
Two Dimensions ◽  
Reaction Diffusion System ◽  
Diffusion System

scholarly journals A New Approach and Solution Technique to Solve Time Fractional Nonlinear Reaction-Diffusion Equations

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Inci Cilingir Sungu ◽  
Huseyin Demir
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Time Stepping ◽  
Nonlinear Reaction ◽  
Differential Transform ◽  
Generalized Differential

A new application of the hybrid generalized differential transform and finite difference method is proposed by solving time fractional nonlinear reaction-diffusion equations. This method is a combination of the multi-time-stepping temporal generalized differential transform and the spatial finite difference methods. The procedure first converts the time-evolutionary equations into Poisson equations which are then solved using the central difference method. The temporal differential transform method as used in the paper takes care of stability and the finite difference method on the resulting equation results in a system of diagonally dominant linear algebraic equations. The Gauss-Seidel iterative procedure then used to solve the linear system thus has assured convergence. To have optimized convergence rate, numerical experiments were done by using a combination of factors involving multi-time-stepping, spatial step size, and degree of the polynomial fit in time. It is shown that the hybrid technique is reliable, accurate, and easy to apply.

scholarly journals Finite‐difference synthetic acoustic logs

Geophysics ◽  
1985 ◽  
Vol 50 (10) ◽  
pp. 1588-1609 ◽  
Author(s):  
R. A. Stephen ◽  
F. Cardo‐Casas ◽  
C. H. Cheng
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Two Dimensions ◽  
Body Waves ◽  
Head Wave ◽  
Stoneley Wave ◽  
Sharp Interface Model ◽  
Head Waves ◽  
The Finite Difference Method

The finite‐difference method is a powerful technique for studying the propagation of elastic waves in boreholes. Even for the simple case of an open borehole with vertical homogeneity, the snapshot format of the method displays clearly the interaction between the borehole and the rock, and the origin and evolution of phases. We present an outline of the finite‐difference method applied to the acoustic logging problem, including a boundary condition formulation for liquid‐solid cylindrical interfaces which is correct to second order in the space increments. Absorbing boundaries based on the formulations of Reynolds (1978) and Clayton and Engquist (1977) were used to reduce reflections from the grid boundaries. Results for a vertically homogeneous sharp interface model are compared with the discrete‐wavenumber method and excellent agreement is obtained. The technique is also demonstrated by considering sharp and continuous transitions (damaged zones) at the borehole wall and by considering the effects of washouts and horizontal fissures on acoustic logs. The latter two cases are examples of wave propagation in media with properties which vary in two dimensions. For the models considered, amplitudes of head waves and head wave multiples (leaky PL modes) are frequently enhanced by washouts. The compressional body waves are less affected by the washouts and horizontal fissures than the guided Stoneley waves which are reflected and only partially transmitted at changes in borehole radius. Amplitude changes of up to 6 dB are observed in the compressional wave due to the borehole deformation. For the Stoneley wave, borehole deformations can cause changes in amplitude of 20 dB and dramatic changes in waveform.

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