scholarly journals A Linear Implicit Finite Difference Discretization of the Schrödinger–Hirota Equation

2018 ◽  
Vol 77 (1) ◽  
pp. 634-656 ◽  
Author(s):  
Georgios E. Zouraris
Keyword(s):  
Finite Difference ◽  
Hirota Equation ◽  
Finite Difference Discretization ◽  
Implicit Finite Difference ◽  
Difference Discretization

An implicit finite-difference operator for the Helmholtz equation

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. T97-T107 ◽  
Author(s):  
Chunlei Chu ◽  
Paul L. Stoffa
Keyword(s):  
Finite Difference ◽  
Helmholtz Equation ◽  
Frequency Domain ◽  
Difference Operator ◽  
Computational Cost ◽  
Finite Difference Discretization ◽  
Implicit Finite Difference ◽  
Explicit Finite Difference ◽  
The One ◽  
Difference Discretization

We have developed an implicit finite-difference operator for the Laplacian and applied it to solving the Helmholtz equation for computing the seismic responses in the frequency domain. This implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost, compared with the corresponding conventional explicit finite-difference scheme. We achieved this by taking advantage of the inherently implicit nature of the Helmholtz equation and merging together the two linear systems: one from the implicit finite-difference discretization of the Laplacian and the other from the discretization of the Helmholtz equation itself. The end result of this simple yet important merging manipulation is a single linear system, similar to the one resulting from the conventional explicit finite-difference discretizations, without involving any differentiation matrix inversions. We analyzed grid dispersions of the discrete Helmholtz equation to show the accuracy of this implicit finite-difference operator and used two numerical examples to demonstrate its efficiency. Our method can be extended to solve other frequency domain wave simulation problems straightforwardly.

LEARNING IN RECURRENT FINITE DIFFERENCE NETWORKS

1995 ◽  
Vol 06 (03) ◽  
pp. 249-256 ◽  
Author(s):  
FU-SHENG TSUNG ◽  
GARRISON W. COTTRELL
Keyword(s):  
Finite Difference ◽  
Time Delays ◽  
Learning Algorithm ◽  
Time Constants ◽  
Learning Time ◽  
Discrete Algorithms ◽  
Finite Difference Discretization ◽  
Difference Discretization ◽  
Discrete Networks ◽  
Period Of Oscillation

A recurrent learning algorithm based on a finite difference discretization of continuous equations for neural networks is derived. This algorithm has the simplicity of discrete algorithms while retaining some essential characteristics of the continuous equations. In discrete networks learning smooth oscillations is difficult if the period of oscillation is too large. The network either grossly distorts the waveforms or is unable to learn at all. We show how the finite difference formulation can explain and overcome this problem. Formulas for learning time constants and time delays in this framework are also presented.

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