• 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.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.27 no.9
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• pp.1327-1334
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• 2003

Flow characteristics of the fluid laden with many particles in the two-dimensional channel are investigated using the Navier-Stokes equations coupled with the equation of motion of particles by direct numerical simulation. A four-step fractional step method with Crank-Nicolson scheme and ALE technique is used for P2P1 mixed finite element method. The motion and distribution of particles in the fluid is virtually described as a result of direct numerical simulation and the increase of viscosity is compared with theoretical equations. The effect of channel height on the relative viscosity and the tubular pinch effect are discussed.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.24 no.8 s.179
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• pp.2042-2049
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• 2000

This paper is concerned with the comparison of forging characteristics between forward and backward processes, through the three-dimensional finite element simulation, for the aluminum powder forging of engine pistons. Starting from the theoretical formulation of velocity and temperature fields in the sintered preform during the process, we examine the comparative distributions of relative density, effective stress and temperature as well as the variations of total forging load and total volume reduction. Through the comparative results, we find out that the forward method provides better forging characteristics than the backward method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.4
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• pp.337-350
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• 2014

In this paper, we consider discontinuous Galerkin finite element methods with interior penalty term to approximate the solution of nonlinear parabolic problems with mixed boundary conditions. We construct the finite element spaces of the piecewise polynomials on which we define fully discrete discontinuous Galerkin approximations using the Crank-Nicolson method. To analyze the error estimates, we construct an appropriate projection which allows us to obtain the optimal order of a priori ${\ell}^{\infty}(L^2)$ error estimates of discontinuous Galerkin approximations in both spatial and temporal directions.

• 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.

• Journal of Korea Water Resources Association
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• v.45 no.2
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• pp.179-189
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• 2012

To simulate the transport of pollutant, a numeric model for the advection-diffusion equation with the decay term is developed. This is finite-difference model using the implicit method (with the weight factor ${\alpha}$) and Gauss-Seidel SOR(successive over-relaxation). This model is compared to the analytical solutions (of simpler dimensional or boundary conditions), and in the condition of Peclet number < 5~20, the result shows stable condition, and Crank-Nicolson method (${\alpha}$=0.5) shows the more accurate results than fully-implicit method (${\alpha}$=1). The mass of advection, diffusion and decay is calculated and the error of mass balance is less than 3%. This model can evaluate the 3-D concentrations of the advection-diffusion and decay problems, but this model uses only the finite-difference method with the fixd grid system, so it can be effectively used in the problems with small Peclet numbers like the pollutant transport in groundwater.

• 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.

• Journal of Korea Water Resources Association
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• v.45 no.1
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• pp.39-52
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• 2012

Surface-subsurface interactions are an intrinsic component of the hydrologic response within a watershed. In general, these interactions are considered to be one of the most difficult areas of the discipline, particularly for the modeler who intends simulate the dynamic relations between these two major domains of the hydrological cycle. In essence, one major complexity is the spatial and temporal variations in the dynamically interacting system behavior. The proper simulation of these variations requires the need for providing an appropriate coupling mechanism between the surface and subsurface components of the system. In this study, an approach for modelling surface-subsurface flow and transport in a fully intergrated way is presented. The model uses the 2-dimensional diffusion wave equation for sheet surface water flow, and the Boussinesq equation with the Darcy's law and Dupuit-Forchheimer's assumption for variably saturated subsurface water flow. The coupled system of equations governing surface and subsurface flows is discretized using the finite volume method with central differencing in space and the Crank-Nicolson method in time. The interactions between surface and subsurface flows are considered mass balance based on the continuity conditions of pressure head and exchange flux. The major module consists of four sub-module (SUBFA, SFA, IA and NS module) is developed.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.22 no.6
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• pp.387-396
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• 2010

A 3-D numeric model for the confined transient flow under sea ground have been developed. This is FDM model using Gauss-Seidel SOR (successive over-relaxation). This model shows the similar head distribution pattern to Theis analytic solution and MODFLOW simulation. The input flowrate to the aquifer and discharge of well have been compared. And it have been found that mass balance is influenced by the weight factor ${\alpha}$, i.e. fullyimplicit method (${\alpha}$=1) shows 5% error, but when ${\alpha}$ becomes to 0.5(Crank and Nicolson method) the mass balance becomes worse and the model result diverges. And the convergency of the model is not much different when $\lambda$ (over-relaxation factor)=0.8~1.5, but when $\lambda$>1.5, the model result diverges. The test-run shows that the well discharge becomes smaller when another well is near. This model can cover the isotropy$(Kx{\neq}Ky{\neq}Kz)$ and inhomogeneity, and can be used for the selection of well site, discharge calculation, and head prediction in case of the artificial recharge etc.

• Journal of Theoretical and Applied Mechanics
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• v.3 no.1
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• pp.45-65
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• 2002

A constitutive model on oorthotropic thermo-elasto-viscoplasticity for fiber-reinforced composite materials Is illustrated, and their thermomechanical responses are predicted with the fully-coupled finite element formulation. The unmixing-mixing scheme can be adopted with the multipartite matrix method as the constitutive model. Basic assumptions based upon the composite micromechanics are postulated, and the strain components of thermal expansion due to temperature change are included In the formulation. Also. more than two sets of mechanical variables, which represent the deformation states of multipartite matrix can be introduced arbitrarily. In particular, the unmixing-mixing scheme can be used with any well-known isotropic viscoplastic theory of the matrix material. The scheme unnecessitates the complex processes for developing an orthotropic viscoplastic theory. The governing equations based on fully-coupled thermomechanics are derived with constitutive arrangement by the unmixing-mixing concept. By considering some auxiliary conditions, the Initial-boundary value problem Is completely set up. As a tool of numerical analyses, the finite element method Is used with isoparametric Interpolation fer the displacement and the temperature fields. The equation of mutton and the energy conservation equation are spatially discretized, and then the time marching techniques such as the Newmark method and the Crank-Nicolson technique are applied. To solve the ultimate nonlinear simultaneous equations, a successive iteration algorithm is constructed with subincrementing technique. As a numerical study, a series of analyses are performed with the main focus on the thermomechanical coupling effect in composite materials. The progress of viscoplastic deformation, the stress-strain relation, and the temperature History are careful1y examined when composite laminates are subjected to repeated cyclic loading.