• Journal of applied mathematics & informatics
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• v.36 no.1_2
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• pp.135-154
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• 2018

In this paper, an almost second order overlapping Schwarz method for singularly perturbed third order convection-diffusion type problem is constructed. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the combination of classical finite difference scheme and central finite difference scheme on a uniform mesh while on the non-layer region we use the midpoint difference scheme on a uniform mesh. It is shown that the numerical approximations which converge in the maximum norm to the exact solution. We proved that, when appropriate subdomains are used, the method produces convergence of second order. Furthermore, it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantages of this method used with the proposed scheme are it reduce iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1279-1292
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• 2009

In this paper, a numerical method that uses standard finite difference scheme defined on Shishkin mesh for a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with a discontinuous source term is presented. An error estimate is derived to show that the method is uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented to illustrate the theoretical results.

• 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.36 no.3_4
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• pp.181-203
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• 2018

In this paper, we have constructed an overlapping Schwarz method for singularly perturbed second order convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region we use the central finite difference scheme on a uniform mesh while on the non-layer region we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations which converge in the maximum norm to the exact solution. When appropriate subdomains are used, the numerical approximations generated from the method are shown to be first order convergent. Furthermore it is shown that, two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantages of this method used with the proposed scheme is it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.

• Journal of computational fluids engineering
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• v.11 no.1 s.32
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• pp.36-42
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• 2006

The existing numerical approximations of convection flux, especially the spatial higher-order difference schemes, in unstructured cell-centered finite volume methods are examined in detail with each other and evaluated with respect to the accuracy through their application to a 2-D benchmark problem. Six higher-order schemes are examined, which include two second-order upwind schemes, two central difference schemes and two hybrid schemes. It is found that the 2nd-order upwind scheme by Mathur and Murthy(1997) and the central difference scheme by Demirdzic and Muzaferija(1995) have more accurate prediction performance than the other higher-order schemes used in unstructured cell-centered finite volume methods.

• Journal of the Korean Crystal Growth and Crystal Technology
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• v.19 no.1
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• pp.39-47
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• 2009

The effects of thermo-solutal convection on mercurous chloride system of ${Hg_2}{Cl_2}$, and Ne during physical vapor transport are numerically investigated for further understanding and insight into essence of transport phenomena, For $10\;K{\le}{\Delta}T{\le}30\;K$, the growth rate slowly increases and, then is decreased gradually until ${\Delta}T$=50 K, The occurrence of this critical point near at ${\Delta}T$=30 K is likely to be due to the effects of thermo-physical properties stronger than the temperature gradient corresponding to driving force for thermal convection. For the range of $10\;Torr{\le}P_B{\le}300\;Torr$, the rate is second order-exponentially decayed with partial pressures of component B, $P_B$. For the range of $5{\le}M_B{\le}200$, the rate is second order-exponentially decayed with a function of molecular weight of component B, $M_B$. Like the case of a partial pressure of component B, the effects of a molecular weight arc: reflected through the binary diffusivity coefficients, which are intimately related with suppressing the convection flow inside the growth enclosure, i,e., transition from convection to diffusion-dominant flow mode as the molecular weight of B increases. The convective mode is near at a ground level, i,e., on earth (1 $g_0$), and the convection is switched to the diffusion mode for $0.1\;g_0{\le}g{\le}10^{-2}g_0$, whereas the diffusion region ranges from $10^{-2}g_0$ up to $10^{-5}g_0$.

• Journal of the Chungcheong Mathematical Society
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• v.20 no.4
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• pp.419-432
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• 2007

Double-diffusive convection inside a triangular porous cavity is studied numerically. Galerkin finite element method is adopted to derive the discrete form of the governing differential equations. The first-order backward Euler scheme is used for temporal discretization with the second-order Adams-Bashforth scheme for the convection terms in the energy and species conservation equations. The Boussinesq-Oberbeck approximation is used to calculate the density dependence on the temperature and concentration fields. A parametric study is performed with the Lewis number, the Rayleigh number, the buoyancy ratio, and the shape of the triangle. The effect of gravity orientation is considered also. Results obtained include the flow, temperature, and concentration fields. The differences induced by varying physical parameters are analyzed and discussed. It is found that the heat transfer rate is sensitive to the shape of the triangles. For the given geometries, buoyancy ratio and Rayleigh numbers are the dominating parameters controlling the heat transfer.

• Journal of the Korean Crystal Growth and Crystal Technology
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• v.20 no.3
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• pp.117-124
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• 2010

For Ar=5, Pr=1.18, Le=0.15, Pe=2.89, Cv=1.06, $P_B$=20 Torr, the effects of impurity $(N_2)$ on thermally and solutally buoyancy-driven convection ($Gr_t=3.46{\times}10^4$ and $Gr_s=6.02{\times}10^5$, respectively) are theoretically investigated for further understanding and insight into an essence of thermo-solutal convection occurring in the vapor phase during the physical vapor transport. For $10K{\leq}{\Delta}T{\leq}50K$, the crystal growth rates are intimately related and linearly proportional to a temperature difference between the source and crystal region which is a driving force for thermally buoyancy-driven convection. Moreover, both the dimensionless Peclet number (Pe) and dimensional maximum velocity magnitudes are directly and linearly proportional to ${\Delta}T$. The growth rate is second order-exponentially decayed for $2{\leq}Ar{\leq}5$. This is related to a finding that the effects of side walls tend to stabilize the thermo-solutal convection in the growth reactor. Finally, the growth rate is found to be first order exponentially decayed for $10{\leq}P_B{\leq}200$ Torr.

• Journal of computational fluids engineering
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• v.18 no.4
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• pp.53-60
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• 2013

The boron transport model has been implemented into the CUPID code to simulate the boron transport phenomena of the PWR. The boron concentration conservation was confirmed through a simulation of a conceptual boron transport problem in which water with a constant inlet boron concentration injected into an inlet of the 2-dimensional vertical flow tube. The step wise boron transport problem showed that the numerical diffusion of the boron concentration can be reduced by the second order convection scheme. In order to assess the adaptability of the developed boron transport model to the realistic situation, the ROCOM test was simulated by using the CUPID implemented with the boron transportation.

• Journal of applied mathematics & informatics
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• v.26 no.5_6
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• pp.1057-1069
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• 2008

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as 'An asymptotic finite element method' for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.