scholarly journals A Fourth Order Finite Difference Method for Numerical Solution of the Goursat Problem

2014 ◽  
Vol 7 (3) ◽  
Author(s):  
P. K. Pandey
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Fourth Order ◽  
Difference Method ◽  
Goursat Problem

FINITE DIFFERENCE METHOD FOR SOLVING THE NONLINEAR DYNAMIC EQUATION OF UNDERWATER TOWED SYSTEM

2014 ◽  
Vol 11 (04) ◽  
pp. 1350060 ◽  
Author(s):  
ZHIJIANG YUAN ◽  
LIANGAN JIN ◽  
WEI CHI ◽  
HENGDOU TIAN
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Difference Method ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Computational Time ◽  
Algebraic Equations ◽  
Nonlinear Dynamic Equation ◽  
Nonlinear Dynamic Equations

A wide body of work exists that describes numerical solution for the nonlinear system of underwater towed system. Many researchers usually divide the tow cable with less number elements for the consideration of computational time. However, this type of installation affects the accuracy of the numerical solution. In this paper, a newly finite difference method for solving the nonlinear dynamic equations of the towed system is developed. The mathematical model of tow cable and towed body are both discretized to nonlinear algebraic equations by center finite difference method. A newly discipline for formulating the nonlinear equations and Jacobian matrix of towed system are proposed. We can solve the nonlinear dynamic equation of underwater towed system quickly by using this discipline, when the size of number elements is large.

scholarly journals the numerical solution of two point boundary value problem for the Helmholtz type equation by finite difference method with non regular step length between nodes

2021 ◽  
Vol 1 (1) ◽  
pp. 18-23
Author(s):  
Pramod Pandey
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Numerical Solution ◽  
Difference Method ◽  
Boundary Value ◽  
Type Equation ◽  
Step Length ◽  
Computational Results ◽  
Order Of Accuracy ◽  
Variable Step

In this article, we have presented a variable step finite difference method for solving second order boundary value problems in ordinary differential equations. We have discussed the convergence and established that proposed has at least cubic order of accuracy. The proposed method tested on several model problems for the numerical solution. The numerical results obtained for these model problems with known / constructed exact solution confirm the theoretical conclusions of the proposed method. The computational results obtained for these model problems suggest that method is efficient and accurate.

scholarly journals A comparison of finite-difference and fourier method calculations of synthetic seismograms

1989 ◽  
Vol 79 (4) ◽  
pp. 1210-1230
Author(s):  
C. R. Daudt ◽  
L. W. Braile ◽  
R. L. Nowack ◽  
C. S. Chiang
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Group Velocity ◽  
Fourth Order ◽  
Difference Method ◽  
Fourier Method ◽  
Velocity Model ◽  
Second Order ◽  
Heterogeneous Model ◽  
Difference Calculation

Abstract The Fourier method, the second-order finite-difference method, and a fourth-order implicit finite-difference method have been tested using analytical phase and group velocity calculations, homogeneous velocity model calculations for disperson analysis, two-dimensional layered-interface calculations, comparisons with the Cagniard-de Hoop method, and calculations for a laterally heterogeneous model. Group velocity rather than phase velocity dispersion calculations are shown to be a more useful aid in predicting the frequency-dependent travel-time errors resulting from grid dispersion, and in establishing criteria for estimating equivalent accuracy between discrete grid methods. Comparison of the Fourier method with the Cagniard-de Hoop method showed that the Fourier method produced accurate seismic traces for a planar interface model even when a relatively coarse grid calculation was used. Computations using an IBM 3083 showed that Fourier method calculations using fourth-order time derivatives can be performed using as little as one-fourth the CPU time of an equivalent second-order finite-difference calculation. The Fourier method required a factor of 20 less computer storage than the equivalent second-order finite-difference calculation. The fourth-order finite-difference method required two-thirds the CPU time and a factor of 4 less computer storage than the second-order calculation. For comparison purposes, equivalent runs were determined by allowing a group velocity error tolerance of 2.5 per cent numerical dispersion for the maximum seismic frequency in each calculation. The Fourier method was also applied to a laterally heterogeneous model consisting of random velocity variations in the lower half-space. Seismograms for the random velocity model resulted in anticipated variations in amplitude with distance, particularly for refracted phases.

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