• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.17 no.3
• /
• pp.197-207
• /
• 2013

The Cahn-Hilliard equation was proposed as a phenomenological model for describing the process of phase separation of a binary alloy. The equation has been applied to many physical applications such as amorphological instability caused by elastic non-equilibrium, image inpainting, two- and three-phase fluid flow, phase separation, flow visualization and the formation of the quantum dots. To solve the Cahn-Hillard equation, many numerical methods have been proposed such as the explicit Euler's, the implicit Euler's, the Crank-Nicolson, the semi-implicit Euler's, the linearly stabilized splitting and the non-linearly stabilized splitting schemes. In this paper, we investigate each scheme in finite-difference schemes by comparing their performances, especially stability and efficiency. Except the explicit Euler's method, we use the fast solver which is called a multigrid method. Our numerical investigation shows that the linearly stabilized stabilized splitting scheme is not unconditionally gradient stable in time unlike the known result. And the Crank-Nicolson scheme is accurate but unstable in time, whereas the non-linearly stabilized splitting scheme has advantage over other schemes on the time step restriction.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.19 no.2
• /
• pp.197-215
• /
• 2015

Modeling water flow in variably saturated, porous media is important in many branches of science and engineering. Highly nonlinear relationships between water content and hydraulic conductivity and soil-water pressure result in very steep wetting fronts causing numerical problems. These include poor efficiency when modeling water infiltration into very dry porous media, and numerical oscillation near a steep wetting front. A one-dimensional finite element formulation is developed for the numerical simulation of variably saturated flow systems. First order backward Euler implicit and second order Crank-Nicolson time discretization schemes are adopted as a solution strategy in this formulation based on Picard and Newton iterative techniques. Five examples are used to investigate the numerical performance of two approaches and the different factors are highlighted that can affect their convergence and efficiency. The first test case deals with sharp moisture front that infiltrates into the soil column. It shows the capability of providing a mass-conservative behavior. Saturated conditions are not developed in the second test case. Involving of dry initial condition and steep wetting front are the main numerical complexity of the third test example. Fourth test case is a rapid infiltration of water from the surface, followed by a period of redistribution of the water due to the dynamic boundary condition. The last one-dimensional test case involves flow into a layered soil with variable initial conditions. The numerical results indicate that the Crank-Nicolson scheme is inefficient compared to fully implicit backward Euler scheme for the layered soil problem but offers same accuracy for the other homogeneous soil cases.

• 한국전산유체공학회:학술대회논문집
• /
• 1999.05a
• /
• pp.186-191
• /
• 1999

An Euler solver is developed to predict accurate aerodynamic data such as lift coefficient, drag coefficient, and moment coefficient. The conservation law form of the compressible Euler equations are used in the generalized curvilinear coordinates system. The Euler solver uses a finite volume method and the second order Roe's flux difference splitting scheme with min-mod flux limiter to calculate the fluxes accurately. An implicit scheme which includes the boundary conditions is implemented to accelerate the convergence rate. The multi-block grid is integrated into the flow solver for complex geometry. The flowfields are analyzed around NACA 0012 airfoil in the cases of $M_{\infty}=0.75,\;\alpha=2.0\;and\;M_{\infty}=0.80,\;\alpha=1.25$. The numerical results are compared with other numerical results from the literature. The final goal of this research is to prepare a robust and an efficient Navier-Stokes solver eventually.

• Journal of applied mathematics & informatics
• /
• v.39 no.5_6
• /
• pp.655-676
• /
• 2021

A parameter uniform numerical scheme is proposed for solving singularly perturbed parabolic partial differential-difference convection-diffusion equations with a small delay and advance parameters in reaction terms and spatial variable. Taylor's series expansion is applied to approximate problems with the delay and advance terms. The resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for the temporal discretization and finite difference method for the spatial discretization on a uniform mesh. The proposed numerical scheme is shown to be an ε-uniformly convergent accurate of the first order in time and second-order in space directions. The efficiency of the scheme is proved by some numerical experiments and by comparing the results with other results. It has been found that the proposed numerical scheme gives a more accurate approximate solution than some available numerical methods in the literature.

• Journal of the Korean Crystal Growth and Crystal Technology
• /
• v.14 no.4
• /
• pp.155-159
• /
• 2004

Numerical analysis which is based on finite element techniques, implicit Euler method and frontal solving algorithm was performed to study the effects of the crucible shape on the temperature of sapphire crystal and the shape of the melt/crystal interface in heat exchanger method. The computer simulation described here and effective to solving the heat transport phenomena with the transition of the interface shape from hemispherical to planar. In the work, various crucibles with differently shaped corners at their bottom are considered to improve the deflection of the melt/crystal interface. The shape of the crucible should be considered as one of the variables for the process optimization.

• 한국전산유체공학회:학술대회논문집
• /
• 2003.10a
• /
• pp.23-25
• /
• 2003

• 한국전산유체공학회:학술대회논문집
• /
• 2006.05a
• /
• pp.385-385
• /
• 2006

• Journal of the Korea Computer Graphics Society
• /
• v.5 no.1
• /
• pp.27-33
• /
• 1999

In this paper, we propose an efficient technique for the animation of flexible thin objects. Mass-spring model was employed to represent the flexible objects. Till now, many techniques have used the mass-spring model to generate plausible animation of soft objects. A straight-forward approach to the animation with mass-spring model is explicit Euler method, but the explicit Euler method has serious disadvantage that it suffers from 'instability problem'. The implicit integration method is a possible solution to overcome the instability problem. However, the most critical flaw of the implicit method is that it involves a large linear system. This paper presents a fast animation technique for mass-spring model with approximated implicit method. The proposed technique stably updates the state of n mass-points in O(n) time when the number of total springs are O(n). We also consider the interaction of the flexible object and air in order to generate plausible results.

• 한국전산유체공학회:학술대회논문집
• /
• 1995.10a
• /
• pp.175-181
• /
• 1995

Three-Dimensional Euler equations are solved numerically for the analysis of contraction flows in wind or water tunnels. A second-order finite difference method is used for the spatial discretization on the nonstaggered grid system and the 4-stage Runge-Kutta scheme for the numerical integration in time. In order to speed up the convergence, the local time stepping and the implicit residual-averaging schemes are introduced. The pressure field is obtained by solving the pressure-Poisson equation with the Neumann boundary condition. For the evaluation of the present Euler solver, numerical computations are carried out for the various contraction geometries, one of which was adopted in the Large Cavitation Channel for the U.S. Navy. The comparison of the computational results with the available experimental data shows good agreements.

• 한국전산유체공학회:학술대회논문집
• /
• 2003.08a
• /
• pp.49-55
• /
• 2003

A comprehensive study has been made for the investigation of the convergence characteristics of the LU scheme for the Euler equations using von Neumann stability analysis. The stability results indicate that the convergence rate is governed by a specific parameter combination. Based on this insight it is shown that the LU scheme will not suffer convergence deterioration at any grid aspect ration if the local time step is defined using appropriate parameter combination. The numerical results demonstrate that this time step definition gives uniform convergence for grid aspect ratios from one to $1\times10^4$.