• Title/Summary/Keyword: nonlinear difference scheme

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A LINEARIZED FINITE-DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION OF THE NONLINEAR CUBIC SCHRODINGER EQUATION

  • Bratsos, A.G.
    • Journal of applied mathematics & informatics
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    • v.8 no.3
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    • pp.683-691
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    • 2001
  • A linearized finite-difference scheme is used to transform the initial/boundary-value problem associated with the nonlinear Schrodinger equation into a linear algebraic system. This method is developed by replacing the time and the nonlinear term by an appropriate parametric linearized scheme based on Taylor’s expansion. The resulting finite-difference method is analysed for stability and convergence. The results of a number of numerical experiments for the single-soliton wave are given.

A PARAMETRIC SCHEME FOR THE NUMERICAL SOLUTION OF THE BOUSSINESQ EQUATION

  • Bratsos, A.G.
    • Journal of applied mathematics & informatics
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    • v.8 no.1
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    • pp.45-57
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    • 2001
  • A parametric scheme is proposed for the numerical solution of the nonlinear Boussinesq equation. The numerical method is developed by approximating the time and the space partical derivatives by finite-difference re placements and the nonlinear term by an appropriate linearized scheme. The resulting finite-difference method is analyzed for local truncation error and stability. The results of a number of numerical experiments are given for both the single and the double-soliton wave. AMS Mathematics Subject Classification : 65J15, 47H17, 49D15.

A CONSERVATIVE NONLINEAR DIFFERENCE SCHEME FOR THE VISCOUS CAHN-HILLIARD EQUATION

  • Choo, S.M.;Chung, S.K.
    • Journal of applied mathematics & informatics
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    • v.16 no.1_2
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    • pp.53-68
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    • 2004
  • Numerical solutions for the viscous Cahn-Hilliard equation are considered using the Crank-Nicolson type finite difference method which conserves the mass. The corresponding stability and error analysis of the scheme are shown. The decay speeds of the solution in $H^1-norm$ are shown. We also compare the evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation numerically and computationally, which has been given as an open question in Novick-Cohen[13].

A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME FOR THE EXTENDED-FISHER-KOLMOGOROV EQUATION

  • Kadri, Tlili;Omrani, Khaled
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.297-310
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    • 2018
  • In this paper, a nonlinear high-order difference scheme is proposed to solve the Extended-Fisher-Kolmogorov equation. The existence, uniqueness of difference solution and priori estimates are obtained. Furthermore, the convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete $L^{\infty}-norm$. Some numerical examples are given in order to validate the theoretical results.

FINITE DIFFERENCE SCHEMES FOR A GENERALIZED NONLINEAR CALCIUM DIFFUSION EQUATION

  • Choo, S.M.
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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    • pp.1247-1256
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    • 2009
  • Finite difference schemes are considered for a nonlinear $Ca^{2+}$ diffusion equations with stationary and mobile buffers. The scheme inherits mass conservation as for the classical solution. Stability and $L^{\infty}$ error estimates of approximate solutions for the corresponding schemes are obtained. using the extended Lax-Richtmyer equivalence theorem.

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EXISTENCE AND MANN ITERATIVE METHODS OF POSITIVE SOLUTIONS OF FIRST ORDER NONLINEAR NEUTRAL DIFFERENCE EQUATIONS

  • Hao, Jinbiao;Kang, Shin Min
    • Korean Journal of Mathematics
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    • v.18 no.3
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    • pp.299-309
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    • 2010
  • In this paper, we study the first order nonlinear neutral difference equation: $${\Delta}(x(n)+px(n-{\tau}))+f(n,x(n-c),x(n-d))=r(n),\;n{\geq}n_0$$. Using the Banach fixed point theorem, we prove the existence of bounded positive solutions of the equation, suggest Mann iterative schemes of bounded positive solutions, and discuss the error estimates between bounded positive solutions and sequences generated by Mann iterative schemes.

On the Suitability of Centered and Upwind-Biased Compact Difference Schemes for Large Eddy Smulation (I) - Numerical Test - (LES에서 중심 및 상류 컴팩트 차분기법의 적합성에 관하여 (I) - 수치 실험 -)

  • Park, No-Ma;Yoo, Jung-Yul;Choi, Hae-Cheon
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.27 no.7
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    • pp.973-983
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    • 2003
  • The suitability of high-order accurate, centered and upwind-biased compact difference schemes is evaluated for large eddy simulation of turbulent flow. Two turbulent flows are considered: turbulent channel flow at Re = 23000 and flow over a circular cylinder at Re = 3900. The effects of numerical dissipation on the finite differencing and aliasing errors and the subgrid-scale stress are investigated. It is shown through the simulations that compact upwind schemes are not suitable for LES, whereas the fourth order-compact centered scheme is a good candidate for LES provided that proper dealiasing of nonlinear terms is performed. The classical issue on the aliasing error and the treatment of nonlinear terms is revisited with compact difference schemes.

A Note on the Modified Scheme for Nonlinear Shallow-Water Equations (비선형 천수방정식의 보정차분기법)

  • 조용식
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.11 no.4
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    • pp.197-200
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    • 1999
  • An extension of the modified leap-frog scheme is made to solve the nonlinear shallow-water equations. In the extended model. the physical dispersion of the Boussinesq equations is replaced by the numerical dispersion resulted from the leap-frog finite difference scheme. The model is used to simulate propagations of a solitary wave over a constant water depth and a linearly varying water depth. Obtained numerical results are compared with available analytical and other numerical solutions. A reasonable agreement is observed.

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NUMERICAL SOLUTIONS FOR SPACE FRACTIONAL DISPERSION EQUATIONS WITH NONLINEAR SOURCE TERMS

  • Choi, Hong-Won;Chung, Sang-Kwon;Lee, Yoon-Ju
    • Bulletin of the Korean Mathematical Society
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    • v.47 no.6
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    • pp.1225-1234
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    • 2010
  • Numerical solutions for the fractional differential dispersion equations with nonlinear forcing terms are considered. The backward Euler finite difference scheme is applied in order to obtain numerical solutions for the equation. Existence and stability of the approximate solutions are carried out by using the right shifted Grunwald formula for the fractional derivative term in the spatial direction. Error estimate of order $O({\Delta}x+{\Delta}t)$ is obtained in the discrete $L_2$ norm. The method is applied to a linear fractional dispersion equations in order to see the theoretical order of convergence. Numerical results for a nonlinear problem show that the numerical solution approach the solution of classical diffusion equation as fractional order approaches 2.

Analysis Run-up of 1993 Hokkaido Nansei Oki Tsunami (1993년 북해도 남서 외해 지진해일 처오름 해석)

  • Kim Jae-Hong;Son Dea-Hee;Cho Yong-Sik
    • Proceedings of the Korea Water Resources Association Conference
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    • 2005.05b
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    • pp.1063-1067
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    • 2005
  • A second-order accuracy upwind scheme is used to investigate the run-up heights of tsunamis in the East Sea and the predicted results are compared with field observed data and results of a first-order accuracy upwind scheme, In the numerical model, the governing equations solved by the finite difference scheme are the linear shallow-water equations in deep water and nonlinear shallow-water equations in shallow water The target events is 1993 Hokktaido Nansei Oki Tsunami. The predicted results represent reasonably the run-up heights of tsunamis in the East Sea. And, The results of simulation is used to design inundation map.

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