An unconditionally stable alternating segment difference scheme of eight points for the dispersive equation

2006 ◽  
Vol 67 (3) ◽  
pp. 435-447 ◽  
Author(s):  
Wenqia Wang ◽  
Shujun Fu
Keyword(s):  
Difference Scheme ◽  
Dispersive Equation ◽  
Unconditionally Stable

scholarly journals Weak solvability of the unconditionally stable difference scheme for the coupled sine-Gordon system

2020 ◽  
Vol 25 (6) ◽  
pp. 997-1014
Author(s):  
Ozgur Yildirim ◽  
Meltem Uzun
Keyword(s):  
Difference Scheme ◽  
Numerical Method ◽  
Existence And Uniqueness ◽  
Numerical Experiments ◽  
Finite Difference Schemes ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
First Order ◽  
The Difference ◽  
Weak Solvability

In this paper, we study the existence and uniqueness of weak solution for the system of finite difference schemes for coupled sine-Gordon equations. A novel first order of accuracy unconditionally stable difference scheme is considered. The variational method also known as the energy method is applied to prove unique weak solvability.We also present a new unified numerical method for the approximate solution of this problem by combining the difference scheme and the fixed point iteration. A test problem is considered, and results of numerical experiments are presented with error analysis to verify the accuracy of the proposed numerical method.

scholarly journals Generation and Analysis of a New Implicit Difference Scheme for the Korteweg-de Vries Equation

2018 ◽  
Vol 173 ◽  
pp. 03006
Author(s):  
Yuri Blinkov ◽  
Vladimir Gerdt ◽  
Konstantin Marinov
Keyword(s):  
Difference Scheme ◽  
Vries Equation ◽  
Implicit Difference Scheme ◽  
Order Of Accuracy ◽  
Unconditionally Stable ◽  
Algorithmic Approach ◽  
De Vries ◽  
Volume Method ◽  
Modified Equation ◽  
Two Parameter

In this paper we apply our computer algebra based algorithmic approach to construct a new finite difference scheme for the two-parameter form of the Korteweg-de Vries equation. The approach combines the finite volume method, numerical integration and difference elimination. Being implicit, the obtained scheme is consistent and unconditionally stable. The modified equation for the scheme shows that its accuracy is of the second order in each of the mesh sizes. Using exact one-soliton solution, we compare the numerical behavior of the scheme with that of the other two schemes known in the literature and having the same order of accuracy. The comparison reveals numerical superiority of our scheme.

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.

scholarly journals An Unconditionally Stable Parallel Difference Scheme for Telegraph Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
A. Borhanifar ◽  
Reza Abazari
Keyword(s):  
Difference Scheme ◽  
Domain Decomposition ◽  
Numerical Experiments ◽  
Telegraph Equation ◽  
Unconditional Stability ◽  
Implicit Method ◽  
Alternating Group ◽  
Unconditionally Stable ◽  
Parallel Difference ◽  
Stability And Accuracy

We use an unconditionally stable parallel difference scheme to solve telegraph equation. This method is based on domain decomposition concept and using asymmetric Saul'yev schemes for internal nodes of each sub-domain and alternating group implicit method for sub-domain's interfacial nodes. This new method has several advantages such as: good parallelism, unconditional stability and better accuracy than original Saul'yev schemes. The details of implementation and proving stability are briefly discussed. Numerical experiments on stability and accuracy are also presented.

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