• Journal of applied mathematics & informatics
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• v.4 no.2
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• pp.487-500
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• 1997

In this paper we introduce a multigrid method for solving the nonliear Urysohn integral equation. The algorithm is derived from a discrete resolvent equation which approximates the continuous resolvent equation of the nonlinear Urysohn integral equa-tion. The algorithm is mathematically equivalent to Atkinson's adap-tive twogrid iteration. But the two are different computationally. We show the convergence of the algorithm and its equivalence to Atkinson's adaptive twogrid iteration. in our numerical example we compare our algorithm to other multigrid methods for solving the nonliear Urysohn integral equation including the nonlinear multigrid nethod introduced by hackbush.

• Journal of computational fluids engineering
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• v.7 no.4
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• pp.19-27
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• 2002

An efficient implicit multigrid method is presented for the Navier-Stokes and k-ω turbulence equations. Freezing and limiting strategies are applied to improve the robustness and convergence of the multigrid method. The eddy viscosity and strongly nonlinear production terms of turbulence are frozen in the coarser grids by passing down the values without update of them. The turbulence equations together with the Navier-Stokes equations, however, are consecutively solved on the coarser grids in a loosely coupled fashion. A simple limit for k is also introduced to circumvent slow-down of convergence. Numerical results for the unseparated and separated transonic airfoil flows show that all computations converge well without any robustness problem and the computing time is reduced to a factor of about 3 by the present multigrid method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.6 no.2
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• pp.9-24
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• 2002

Total Variation(TV) regularization method is effective for reconstructing "blocky", discontinuous images from contaminated image with noise. But TV is represented by highly nonlinear integro-differential equation that is hard to solve. There have been much effort to obtain stable and fast methods. C. Vogel introduced "the Fixed Point Lagged Diffusivity Iteration", which solves the nonlinear equation by linearizing. In this paper, we apply multigrid(MG) method for cell centered finite difference (CCFD) to solve system arise at each step of this fixed point iteration. In numerical simulation, we test various images varying noises and regularization parameter $\alpha$ and smoothness $\beta$ which appear in TV method. Numerical tests show that the parameter ${\beta}$ does not affect the solution if it is sufficiently small. We compute optimal $\alpha$ that minimizes the error with respect to $L^2$ norm and $H^1$ norm and compare reconstructed images.

• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.453-464
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• 2007

A computationally efficient numerical scheme is presented for the phase-field model of two-phase systems for anisotropic interfacial energy. The scheme is solved by using a nonlinear multigrid method. When the coefficient for the anisotropic interfacial energy is sufficiently high, the interface of the system shows corners or missing crystallographic orientations. Numerical simulations with high and low anisotropic coefficients show excellent agreement with exact equilibrium shapes. We also present spinodal decomposition, which shows the robustness of the pro-posed scheme.

• Proceedings of the KSME Conference
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• 2002.08a
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• pp.31-34
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• 2002

An efficient implicit multigrid method is presented for the Navier-Stokes and $k-{\omega}$ turbulence equations. Freezing and limiting strategies are applied to improve the robustness and convergence of the multigrid method. The eddy viscosity and strongly nonlinear production terms of turbulence are frozen in the coarser grids by passing down the values without update of them. The turbulence equations together with the Navier-Stokes equations, however, are consecutively solved on the coarser grids in a loosely coupled fashion. A simple limit for k is also introduced to circumvent slow-down of convergence. Numerical results for the unseparated and separated transonic airfoil flows show that all computations converge well without any robustness problem and the computing time is reduced to a factor of about 3 by the present multigrid method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.18 no.1
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• pp.27-41
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• 2014

In this paper, we present an efficient numerical method for multiphase image segmentation using a multiphase-field model. The method combines the vector-valued Allen-Cahn phase-field equation with initial data fitting terms containing prescribed interface width and fidelity constants. An efficient numerical solution is achieved using the recently developed hybrid operator splitting method for the vector-valued Allen-Cahn phase-field equation. We split the modified vector-valued Allen-Cahn equation into a nonlinear equation and a linear diffusion equation with a source term. The linear diffusion equation is discretized using an implicit scheme and the resulting implicit discrete system of equations is solved by a multigrid method. The nonlinear equation is solved semi-analytically using a closed-form solution. And by treating the source term of the linear diffusion equation explicitly, we solve the modified vector-valued Allen-Cahn equation in a decoupled way. By decoupling the governing equation, we can speed up the segmentation process with multiple phases. We perform some characteristic numerical experiments for multiphase image segmentation.

• Communications of the Korean Mathematical Society
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• v.20 no.1
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• pp.179-193
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• 2005

We present results of fully nonlinear time-dependent simulations of a thin liquid film flowing up an inclined plane. Equations of the type $h_t+f_y(h) = -{\in}^3{\nabla}{\cdot}(M(h){\nabla}{\triangle}h)$ arise in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, t) is the fluid film height. A hybrid scheme is constructed for the solution of two-dimensional higher-order nonlinear diffusion equations. Problems in the fluid dynamics of thin films are solved to demonstrate the accuracy and effectiveness of the hybrid scheme.

• Journal of computational fluids engineering
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• v.9 no.1
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• pp.34-40
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• 2004

Recently with growth in the capability of super computers and Parallel computers, shape design optimization is becoming easible for real problems. Also, Computational Fluid Dynamics(CFD) techniques have been improved for higher reliability and higher accuracy. In the shape design optimization, analysis solvers and optimization schemes are essential. In this work, the Roe's 2nd-order Upwind TVD scheme and DADI time march with multigrid were used for the flow solution with the Euler equation and FDM(Finite Differenciation Method), GA(Genetic Algorithm) and Kriging were used for the design optimization. Kriging were applied to 2-D airfoil design optimization and compared with FDM and GA's results. When Kriging is applied to the nonlinear problems, satisfactory results were obtained. From the result design optimization by Kriging method appeared as good as other methods.