• Coupled systems mechanics
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• 제8권2호
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• pp.129-146
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• 2019

The main goal of this work is to develop a numerical simulator to study an isothermal single-phase two-component flow in a naturally fractured oil reservoir, taking into account advection and diffusion effects. We use the Peng-Robinson equation of state with a volume translation to evaluate the properties of the components, and the discretization of the governing partial differential equations is carried out using the Finite Difference Method, along with implicit and first-order upwind schemes. This process leads to a coupled non-linear algebraic system for the unknowns pressure and molar fractions. After a linearization and the use of an operator splitting, the Conjugate Gradient and Bi-conjugated Gradient Stabilized methods are then used to solve two algebraic subsystems, one for the pressure and another for the molar fraction. We studied the effects of fractures in both the flow field and mass transport, as well as in computing time, and the results show that the fractures affect, as expected, the flow creating a thin preferential path for the mass transport.

• Nuclear Engineering and Technology
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• 제54권1호
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• pp.49-60
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• 2022

Motivated by the high-resolution properties of high-order Weighted Essentially Non-Oscillatory (WENO) and flux limiter (FL) for steep-gradient problems and the robust convergence of Jacobian-free Newton-Krylov (JFNK) methods for nonlinear systems, the preconditioned JFNK fully implicit high-order WENO and FL schemes are proposed to solve the transient two-phase two-fluid models. Specially, the second-order fully-implicit BDF2 is used for the temporal operator and then the third-order WENO schemes and various flux limiters can be adopted to discrete the spatial operator. For the sake of the generalization of the finite-difference-based preconditioning acceleration methods and the excellent convergence to solve the complicated and various operational conditions, the random vector instead of the initial condition is skillfully chosen as the solving variables to obtain better sparsity pattern or more positions of non-zero elements in this paper. Finally, the WENO_JFNK and FL_JFNK codes are developed and then the two-phase steep-gradient problem, phase appearance/disappearance problem, U-tube problem and linear advection problem are tested to analyze the convergence, computational cost and efficiency in detailed. Numerical results show that WENO_JFNK and FL_JFNK can significantly reduce numerical diffusion and obtain better solutions than traditional methods. WENO_JFNK gives more stable and accurate solutions than FL_JFNK for the test problems and the proposed finite-difference-based preconditioning acceleration methods based on the random vector can significantly improve the convergence speed and efficiency.

• 대한수학회보
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• 제36권1호
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• pp.63-78
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• 1999

Let $\mu$(x) be an increasing function on the real line with finite moments of all oeders. We show that for any linear operator T on the space of polynomials and any interger n $\geq$ 0, there is a constant $\gamma n(T)\geq0$, independent of p(x), such that $\parallel T_p\parallel\leq\gamma n(T)\parallel P\parallel$, for any polynomial p(x) of degree $\leq$ n, where We find a formular for the best possible value $\Gamma_n(T)\;of\;\gamma n(T)$ and estimations for $\Gamma_n(T)$. We also give several illustrating examples when T is a differentiation or a difference operator and $d\mu$(x) is an orthogonalizing measure for classical or discrete orthogonal polynomials.

• 지구물리와물리탐사
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• 제1권2호
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• pp.116-126
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• 1998

본 논문은 파동장의 심도방향으로의 외삽(extrapolation) 을 사용한 데이터밍 기법을 소개한다. 개발된 기법은 phase-shift, split-step, 또는 유한차분과 같은 다양한 파동장 외삽기법들을 사용할 수 있다. 데이터밍 알고리즘을 유도하기 위해, 우선 평면에 정의 되어 있는 파동장을 임의의 굴곡을 갖는 면으로 외삽을 수행하는 모델링 연산자를 대수학적으로 구한 후, 본 모델링 연산자에 어드조인트(adjoint)관계에 있는 연산자를 대수학적으로 구하여 데이터밍 연산자를 얻었다. 다양한 외삽방법을 사용한 데이터밍 알고리즘의 실험에서 매우 만족스러운 결과를 얻을 수 있었다.

• 지구물리와물리탐사
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• 제10권2호
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• pp.111-123
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• 2007

본 논문에서는 파동장 외삽(wavefield extrapolation)의 방향을 단순히 역시간(reverse time)으로 하여 적용하는 기존의 역시간 구조보정법(reverse time migration method)이 아닌, 묵시적으로 가정된 순방향 모델링(forward modeling) 연산자에 대한 정확한 어드조인트(adjoint) 연산자로서의 역시간 구조보정 연산자를 유도한다. 어드조인트 연산자를 얻는 방법으로는 우선 해당하는 순방향 연산자를 명시적인 행렬식의 형태로 정의하고 이에 대한 전치행렬식을 구한 후, 이러한 전치행렬식에 해당하는 연산자를 정의하는 접근법을 사용하였다. 정확한 어드조인트 관계에 있는 역시간 구조보정 연산자는 기존의 역시간 구조보정 연산자와 마찬가지로 구조보정의 목적으로 사용될 수 있을 뿐 아니라, 최소자승 구조보정(Least-squares migration)과 같은 역산을 통해서 지하구조 영상화를 할 때 필요로 하는 어드조인트 연산자를 정확하게 구현 할 수 있어 보다 정확한 역산 결과를 얻게 해준다.

• 한국지구과학회지
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• 제36권5호
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• pp.476-483
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• 2015

The high-order Laplacian-type filter, which is capable of providing isotropic and sharp cut-off filtering on the spherical domain, is essential in processing geophysical data. In this study, a spherical high-order filter was designed by combining the Fourier method with finite difference-method in the longitude and latitude, respectively. The regular grid system was employed in the filter, which has uniform angular spacing including the poles. The singularity at poles was eliminated by incorporating variable transforms and continuity-matching boundary conditions across poles. The high-order filter was assessed using the Rossby-Haurwitz wave, the observed geopotential, and observed wind field. The performance of the filter was found comparable to the filter based on the Galerkin procedure. The filter, employing the finite difference method, can be designed to give any target order of accuracy, which is an important advantage being unavailable in other methods. The computational complexity is represented with 2n-1 diagonal matrices solver with n being the target order of accuracy. Along with the availability of arbitrary target-order, it is also advantageous that the filter can adopt the reduced grid to increase computational efficiency.

• 한국수자원학회:학술대회논문집
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• 한국수자원학회 2006년도 학술발표회 논문집
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• pp.22-27
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• 2006

This paper describes the development of the stream-tube based dispersion model for modeling contaminant transport in open channels. The operator-splitting technique is employed to separate the 2D contaminant transport equation into the pure advection and pure dispersion equations. Then the total variation diminishing (TVD) schemes are combined with the second-order Lax-Wendroff and third-order QUICKEST explicit finite difference schemes respectively to solve the pure advection equation in order to prevent the occurrence of numerical oscillations. Due to various limiters owning different features, the numerical tests for 1D pure advection and 2D dispersion are conducted to evaluate the performance of different TVD schemes firstly, then the TVD schemes are applied to experimental data for simulating the 2D mixing in a straight trapezoidal channel to test the model capability. Both the numerical tests and model application show that the TVD schemes are very competent for solving the advection-dominated transport problems.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제18권1호
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• pp.61-74
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• 2014

This paper presents accurate and efficient numerical methods for calculating the sensitivities of two-asset European options, the Greeks. The Greeks are important financial instruments in management of economic value at risk due to changing market conditions. The option pricing model is based on the Black-Scholes partial differential equation. The model is discretized by using a finite difference method and resulting discrete equations are solved by means of an operator splitting method. For Delta, Gamma, and Theta, we investigate the effect of high-order discretizations. For Rho and Vega, we develop an accurate and robust automatic algorithm for finding an optimal value. A cash-or-nothing option is taken to demonstrate the performance of the proposed algorithm for calculating the Greeks. The results show that the new treatment gives automatic and robust calculations for the Greeks.

• East Asian mathematical journal
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• 제35권1호
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• pp.99-108
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• 2019

In this paper, we introduce a numerical approach to solve heat-diffusion equation with discontinuous diffusion coefficients in the three dimensional rectangular domain. First, we study the support operator method and suggest a new method, the continuous velocity method. Further, we apply both methods to a diffusion process for neurotransmitter release in an individual synapse and compare their results.

• 한국환경과학회지
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• 제11권9호
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• pp.907-914
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• 2002

Various Eulerian-Lagrangian models for the one-dimensional longitudinal dispersion equation in nonuniform flow were studied comparatively. In the models studied, the transport equation was decoupled into two component parts by the operator-splitting approach; one part is governing advection and the other is governing dispersion. The advection equation has been solved by using the method of characteristics following fluid particles along the characteristic line and the results were interpolated onto an Eulerian grid on which the dispersion equation was solved by Crank-Nicholson type finite difference method. In the solution of the advection equation, Lagrange fifth, cubic spline, Hermite third and fifth interpolating polynomials were tested by numerical experiment and theoretical error analysis. Among these, Hermite interpolating polynomials are generally superior to Lagrange and cubic spline interpolating polynomials in reducing both dissipation and dispersion errors.