• Honam Mathematical Journal
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• v.42 no.1
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• pp.17-35
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• 2020

In this paper we introduce and consider a new class of variational inequalities with four operators. This class is called the extended general mixed quasi variational inequality. We show that the extended general mixed quasi variational inequality is equivalent to the fixed point problem. We use this alternative equivalent formulation to discuss the existence of a solution of extended general mixed quasi variational inequality and also develop several iterative methods for solving extended general mixed quasi variational inequality and its variant forms. We consider the convergence analysis of the proposed iterative methods under appropriate conditions. We also introduce a new class of resolvent equation, which is called the extended general implicit resolvent equation and establish an equivalent relation between the extended general implicit resolvent equation and the extended general mixed quasi variational inequality. Some special cases are also discussed.

• Structural Engineering and Mechanics
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• v.75 no.5
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• pp.541-557
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• 2020

A novel family of controllable, dissipative structure-dependent integration methods is derived from an eigen-based theory, where the concept of the eigenmode can give a solid theoretical basis for the feasibility of this type of integration methods. In fact, the concepts of eigen-decomposition and modal superposition are involved in solving a multiple degree of freedom system. The total solution of a coupled equation of motion consists of each modal solution of the uncoupled equation of motion. Hence, an eigen-dependent integration method is proposed to solve each modal equation of motion and an approximate solution can be yielded via modal superposition with only the first few modes of interest for inertial problems. All the eigen-dependent integration methods combine to form a structure-dependent integration method. Some key assumptions and new techniques are combined to successfully develop this family of integration methods. In addition, this family of integration methods can be either explicitly or implicitly implemented. Except for stability property, both explicit and implicit implementations have almost the same numerical properties. An explicit implementation is more computationally efficient than for an implicit implementation since it can combine unconditional stability and explicit formulation simultaneously. As a result, an explicit implementation is preferred over an implicit implementation. This family of integration methods can have the same numerical properties as those of the WBZ-α method for linear elastic systems. Besides, its stability and accuracy performance for solving nonlinear systems is also almost the same as those of the WBZ-α method. It is evident from numerical experiments that an explicit implementation of this family of integration methods can save many computational efforts when compared to conventional implicit methods, such as the WBZ-α method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.3
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• pp.197-207
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• 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 computational fluids engineering
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• v.16 no.4
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• pp.72-83
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• 2011

A high-order discontinuous Galerkin method for the two-dimensional compressible Navier-Stokes equations was developed on unstructured triangular meshes. For this purpose, the BR2 methd(the second Bassi and Rebay discretization) was adopted for space discretization and an implicit Euler backward method was used for time integration. Numerical tests were conducted to estimate the convergence order of the numerical solutions of the Poiseuille flow for which analytic solutions are available for comparison. Also, the flows around a flat plate, a 2-D circular cylinder, and an NACA0012 airfoil were numerically simulated. The numerical results showed that the present implicit discontinuous Galerkin method is an efficient method to obtain very accurate numerical solutions of the compressible Navier-Stokes equations on unstructured meshes.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.25 no.4
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• pp.224-261
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• 2021

The interest in implicit-explicit (IMEX) integration methods has emerged as an alternative for dealing in a computationally cost-effective way with stiff ordinary differential equations arising from practical modeling problems. In this paper, we introduce implicit-explicit second derivative linear multi-step methods (IMEX SDLMM) with error control. The proposed IMEX SDLMM is based on second derivative backward differentiation formulas (SDBDF) and recursive SDBDF. The IMEX second derivative schemes are constructed with order p ranging from p = 1 to 8. The methods are numerically validated on well-known stiff equations.

• Journal of KIISE:Computer Systems and Theory
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• v.32 no.9
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• pp.445-456
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• 2005

We propose a fast and accurate fluid solver of the Wavier-Stokes equations for the physics-based fluid simulations. Our method utilizes the solution of the Stokes equation as an initial guess for the velocity of the nonlinear term in the Wavier-Stokes equations. By guessing the initial velocity close to the exact solution of the given nonlinear differential equations, we can develop remarkably accurate and stable fluid solver. Our solver is based on the implicit scheme of finite difference methods, that makes it work well for large time steps. Since we employ the ADI method, our solver is also fast and has a uniform computation time. The experimental results show that our solver is excellent for fluids with high Reynolds numbers such as smoke and clouds.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.34 no.3
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• pp.151-157
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• 2021

This paper is devoted to a convergence study of the nonlocal integral operator in peridynamics. The implicit formulation can be an efficient approach to obtain the static/quasi-static solution of crack propagation problems. Implicit methods require constly large-matrix operations. Therefore, convergence is important for improving computational efficiency. When the radial influence function is utilized in the nonlocal integral equation, the fractional Laplacian integral equation is obtained. It has been mathematically proved that the condition number of the system matrix is affected by the order of the radial influence function and nonlocal horizon size. We formulate the static crack problem with peridynamics and utilize Newton-Raphson methods with a preconditioned conjugate gradient scheme to solve this nonlinear stationary system. The convergence behavior and the computational time for solving the implicit algebraic system have been studied with respect to the order of the radial influence function and nonlocal horizon size.

• Journal of Korean Society of Coastal and Ocean Engineers
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• v.12 no.2
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• pp.70-80
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• 2000

Princeton Ocean Model (POM) is modified to construct a three-dimensional, semi-implicit hydro¬dynamic model with a wetting-and-drying scheme. The model employs semi-implicit treatment of the barotropic pressure gradient terms and the vertical mixing terms in the momentum equations, and the velocity divergence term in the vertically-integrated continuity equation. Such treatment removes the external mode and thus the mode splitting scheme in POM, allowing the semi-implicit model to use a larger time step. Applied to hypothetical systems, both the semi-implicit model and POM give nearly the same results. The semi-implicit model, however, runs approximately 4.4 times faster than POM showing its improved computational efficiency. Applied to a hypothetical system with intertidal flats, POM employing the mode splitting scheme produces noises at the intertidal flats, that propagate into the main channel resulting in unstable current velocities. Despite its larger time step, the semi-implicit model gives stable current velocities both at the intertidal flats and main channel. The semi-implicit model when applied to Kyeonggi Bay gives a good reproduction of the observed tides and tidal currents throughout the modeling domain, demonstrating its prototype applicability.

• Communications of the Korean Mathematical Society
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• v.14 no.4
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• pp.833-842
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• 1999

In this paper, we construct a fully discrete solution of the incompressible Navier Stokes equations using implicit Runge kutta method. We prove the existence of the fully discrete solution.

• East Asian mathematical journal
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• v.29 no.1
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• pp.69-82
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• 2013

In this paper we consider a doubly nonlinear Volterra equation related to the Leray-Lions with a nonsmooth kernel. By exploiting a suitable implicit time-discretization technique we obtain the existence of global strong solution.