• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.15 no.4
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• pp.291-306
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• 2011

We consider the second order Strang splitting method to approximate the solution to the KdV equation. The model equation is split into three sets of initial value problems containing convection and dispersal terms separately. TVD MUSCL or MUSCL scheme is applied to approximate the convection term and the second order centered difference method to approximate the dispersal term. In time stepping, explicit third order Runge-Kutta method is used to the equation containing convection term and implicit Crank-Nicolson method to the equation containing dispersal term to reduce the CFL restriction. Several numerical examples of weakly and strongly dispersive problems, which produce solitons or dispersive shock waves, or may show instabilities of the solution, are presented.

• 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].

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1477-1487
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• 2011

We propose a numerical method for solving Allen-Cahn equation, in both one-dimensional and two-dimensional cases. The new scheme that is explicit, stable, and easy to compute is obtained and the proposed method provides a straightforward and effective way for nonlinear evolution equations.

• The Pure and Applied Mathematics
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• v.13 no.1 s.31
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• pp.19-38
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• 2006

We analyze the spectral collocation approximation for a parabolic partial integrodifferential equations(PIDE) with a weakly singular kernel. The space discretization is based on the spectral collocation method and the time discretization is based on Crank-Nicolson scheme with a graded mesh. We obtain the stability and second order convergence result for fully discrete scheme.

• Advances in Energy Research
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• v.5 no.3
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• pp.255-275
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• 2017

The advantages of using natural circulation (NC) as a cooling system, has prompted the worldwide development to investigate this phenomenon more than before. The interesting application of the NC in low power experimental facilities and research reactors, highlights the obligation of study in these laminar flows. The inherent oscillations of NC between hot source and cold sink in low Grashof numbers necessitates stability analysis of cooling flow with experimental or numerical schemes. For this type of analysis, numerical methods could be implemented to desired mass, momentum and energy equations as an efficient instrument for predicting the behavior of the flow field. In this work, using the explicit, implicit and Crank-Nicolson methods, the fluid flow parameters in a natural circulation experimental test loop are obtained and the sensitivity of solving approaches are discussed. In this way, at first, the steady state and transient results from explicit are obtained and compared with experimental data. The implicit and crank-Nicolson scheme is investigated in next steps and in subsequent this research is focused on the numerical aspects of instability prediction for these schemes. In the following, the assessment of the flow behavior with coarse and fine mesh sizes and time-steps has been reported and the numerical schemes convergence are compared. For more detail research, the natural circulation of fluid was modeled by ANSYS-CFX software and results for the experimental loop are shown. Finally, the stability map for rectangular closed loop was obtained with employing the Nyquist criterion.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.10 no.4
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• pp.141-150
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• 1990

Numerical experiments of sballow water equations are performed under various boundary conditions by finite element method to simulate the circulation in estuaries and coastal areas. Galerkin method is employed to discretize spatial domain, and for time integration, finite difference method (Crank-Nicolson scheme) is used. This method is tested in five problems, in which first four cases have analytic solutions. The computed values are well in agreement with the analytic solutions in four experiments and the result of the last 2-dimensional ease is resonable. Implicit and two step Lax-Wendroff schemes in time domain are compared, and the results when using four node bilinear and triangular elements are presented. Consequently it takes very long time for complex problems requiring many elements to integrate all the time steps using the implicit schemes. And the explicit scheme requires careful consideration in selecting the time step and the grid size to obtain the desired accuracy.

• Bulletin of the Korean Mathematical Society
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• v.57 no.1
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• pp.219-244
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• 2020

In this paper, we approximate the solution of the coupled nonlinear Schrödinger equations by using a fully discrete finite element scheme based on the standard Galerkin method in space and implicit midpoint discretization in time. The proposed scheme guarantees the conservation of the total mass and the energy. First, a priori error estimates for the fully discrete Galerkin method is derived. Second, the existence of the approximated solution is proved by virtue of the Brouwer fixed point theorem. Moreover, the uniqueness of the solution is shown. Finally, convergence orders of the fully discrete Crank-Nicolson scheme are discussed. The end of the paper is devoted to some numerical experiments.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.623-641
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• 2021

In this paper, numerical treatment of singularly perturbed parabolic delay differential equations is considered. The considered problem have small delay on the spatial variable of the reaction term. To treat the delay term, Taylor series approximation is applied. The resulting singularly perturbed parabolic PDEs is solved using Crank Nicolson method in temporal direction with non-standard finite difference method in spatial direction. A detail stability and convergence analysis of the scheme is given. We proved the uniform convergence of the scheme with order of convergence O(N^-1 + (∆t)²), where N is the number of mesh points in spatial discretization and ∆t is mesh length in temporal discretization. Two test examples are used to validate the theoretical results of the scheme.

• Proceedings of the KSME Conference
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• 2001.06e
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• pp.392-397
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• 2001

A numerical study is performed to investigate the interaction between subsonic axial turbine blade boundary layer and periodically oncoming rotor induced wakes. An implicit scheme for solving the compressible Navier-Stokes equation is developed, which adopts a 4th-order compact difference for spatial discretiztion, a 2nd order Crank-Nicolson scheme for temporal discretization and the dynamic eddy viscosity model as the subgrid scale model. The efficiency and the accuracy of the proposed method are verified by applying to some benchmark problems such as laminar cylinder flow, laminar airfoil cascade flow and a transitional flat plate boundary layer flow. Computational results show good agreements with previous experimental and numerical results. Finally, flow through a stator cascade is simulated at $Re = 7.5{\times}10^5$ without free-stream turbulence intensity. The velocity fields and skin friction coefficients in the transitional region show similar trends with previous boundary layer natural transition.

• 한국전산유체공학회:학술대회논문집
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• 2000.05a
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• pp.33-38
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• 2000

A Numerical study of laminar vortex-shedding past a circular cylinder has been performed widely by many researchers. Many factors, such as numerical technique and domain size, number and shape of grid, affected predicting vortex shedding and Strouhal number. In the present study, the effect of convection scheme, time discretization methods and grid dependence were investigated. The present paper presents the finite volume solution of unsteady flow past circular cylinder at Re=200, 400. The Strouhal number was predicted using UDS, CDS, Hybrid, Power-law, LUDS, QUICK scheme for convection term, implicit and crank-nicolson methods for time discretization. The grid dependence was investigated using H-type mesh and O-type mesh. It also studied that the effect of mesh size of the nearest adjacent grid of circular cylinder. The effect of convection scheme is greater than the effect of time discretization on predicting Strouhal. It has been found that the predicted Strouhal number changed with mesh size and shape.