• Title, Summary, Keyword: operator splitting method

### Operator-splitting methods respecting eigenvalue problems for shallow shelf equations with basal drag

• Geiser, Jurgen;Calov, Reinhard
• Coupled systems mechanics
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• v.1 no.4
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• pp.325-343
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• 2012
• We present different numerical methods for solving the shallow shelf equations with basal drag (SSAB). An alternative approach of splitting the SSAB equation into a Laplacian and diagonal shift operator is discussed with respect to the underlying eigenvalue problem. First, we solve the equations using standard methods. Then, the coupled equations are decomposed into operators for membranes stresses, basal shear stress and driving stress. Applying reasonable parameter values, we demonstrate that the operator of the membrane stresses is much stiffer than the operator of the basal shear stress. Here, we could apply a new splitting method, which alternates between the iteration on the membrane-stress operator and the basal-shear operator, with a more frequent iteration on the operator of the membrane stresses. We show that this splitting accelerates and stabilize the computational performance of the numerical method, although an appropriate choice of the standard method used to solve for all operators in one step speeds up the scheme as well.

### HIGHER ORDER OPERATOR SPLITTING FOURIER SPECTRAL METHODS FOR THE ALLEN-CAHN EQUATION

• SHIN, JAEMIN;LEE, HYUN GEUN;LEE, JUNE-YUB
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.21 no.1
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• pp.1-16
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• 2017
• The Allen-Cahn equation is solved numerically by operator splitting Fourier spectral methods. The basic idea of the operator splitting method is to decompose the original problem into sub-equations and compose the approximate solution of the original equation using the solutions of the subproblems. The purpose of this paper is to characterize higher order operator splitting schemes and propose several higher order methods. Unlike the first and the second order methods, each of the heat and the free-energy evolution operators has at least one backward evaluation in higher order methods. We investigate the effect of negative time steps on a general form of third order schemes and suggest three third order methods for better stability and accuracy. Two fourth order methods are also presented. The traveling wave solution and a spinodal decomposition problem are used to demonstrate numerical properties and the order of convergence of the proposed methods.

### COMPARISON OF NUMERICAL METHODS (BI-CGSTAB, OS, MG) FOR THE 2D BLACK-SCHOLES EQUATION

• Jeong, Darae;Kim, Sungki;Choi, Yongho;Hwang, Hyeongseok;Kim, Junseok
• The Pure and Applied Mathematics
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• v.21 no.2
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• pp.129-139
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• 2014
• In this paper, we present a detailed comparison of the performance of the numerical solvers such as the biconjugate gradient stabilized, operator splitting, and multigrid methods for solving the two-dimensional Black-Scholes equation. The equation is discretized by the finite difference method. The computational results demonstrate that the operator splitting method is fastest among these solvers with the same level of accuracy.

### AN OPERATOR SPLITTING METHOD FOR PRICING THE ELS OPTION

• Jeong, Da-Rae;Wee, In-Suk;Kim, Jun-Seok
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.3
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• pp.175-187
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• 2010
• This paper presents the numerical valuation of the two-asset step-down equitylinked securities (ELS) option by using the operator-splitting method (OSM). The ELS is one of the most popular financial options. The value of ELS option can be modeled by a modified Black-Scholes partial differential equation. However, regardless of whether there is a closedform solution, it is difficult and not efficient to evaluate the solution because such a solution would be represented by multiple integrations. Thus, a fast and accurate numerical algorithm is needed to value the price of the ELS option. This paper uses a finite difference method to discretize the governing equation and applies the OSM to solve the resulting discrete equations. The OSM is very robust and accurate in evaluating finite difference discretizations. We provide a detailed numerical algorithm and computational results showing the performance of the method for two underlying asset option pricing problems such as cash-or-nothing and stepdown ELS. Final option value of two-asset step-down ELS is obtained by a weighted average value using probability which is estimated by performing a MC simulation.

### Coherent Control of Autler-Townes Splitting in Photoelectron Spectroscopy: The Effect of Laser Intensity and Laser Envelope

• Qin, Chaochao;Zhai, Hongsheng;Zhang, Xianzhou;Liu, Yufang
• Bulletin of the Korean Chemical Society
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• v.35 no.11
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• pp.3294-3298
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• 2014
• We theoretically investigated the coherent control of Autler-Townes splitting in photoelectron spectroscopy of K2 molecule within an ultrafast laser pulse by solving the time-dependent Schrodinger equation using a quantum wave packet method. It was theoretically shown that we can manipulate the splitting of photoelectron spectroscopy by altering the laser intensity. Furthermore, it was found that the percentages of each peak in photoelectron spectroscopy can be controlled by changing the envelope of the laser pulse.

### RELATIONSHIPS AMONG CHARACTERISTIC FINITE ELEMENT METHODS FOR ADVECTION-DIFFUSION PROBLEMS

• CHEN, ZHANGXIN
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.6 no.1
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• pp.1-15
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• 2002
• Advection-dominated transport problems possess difficulties in the design of numerical methods for solving them. Because of the hyperbolic nature of advective transport, many characteristic numerical methods have been developed such as the classical characteristic method, the Eulerian-Lagrangian method, the transport diffusion method, the modified method of characteristics, the operator splitting method, the Eulerian-Lagrangian localized adjoint method, the characteristic mixed method, and the Eulerian-Lagrangian mixed discontinuous method. In this paper relationships among these characteristic methods are examined. In particular, we show that these sometimes diverse methods can be given a unified formulation. This paper focuses on characteristic finite element methods. Similar examination can be presented for characteristic finite difference methods.

### Suggestion for a splitting technique of the square-root operator of three dimensional acoustic parabolic equation based on two variable rational approximant with a factored denominator (인수분해 된 분모를 갖는 두 변수 유리함수 근사에 기반한 3차원 음향 포물선 방정식 제곱근 연산자의 분할기법 제안)

• Lee, Keunhwa
• The Journal of the Acoustical Society of Korea
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• v.36 no.1
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• pp.1-11
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• 2017
• In this study, novel approximate form of the square-root operator of three dimensional acoustic Parabolic Equation (3D PE) is proposed using a rational approximant for two variables. This form has two advantages in comparison with existing approximation studies of the square-root operator. One is the wide-angle capability. The proposed form has wider angle accuracy to the inclination angle of ${\pm}62^{\circ}$ from the range axis of 3D PE at the bearing angle of $45^{\circ}$, which is approximately three times the angle limit of the existing 3D PE algorithm. Another is that the denominator of our approximate form can be expressed into the product of one-dimensional operators for depth and cross-range. Such a splitting form is very preferable in the numerical analysis in that the 3D PE can be easily transformed into the tridiagonal matrix equation. To confirm the capability of the proposed approximate form, comparative study of other approximation methods is conducted based on the phase error analysis, and the proposed method shows best performance.

### COMPARATIVE STUDY OF NUMERICAL ALGORITHMS FOR THE ARITHMETIC ASIAN OPTION

• WANG, JIAN;BAN, JUNGYUP;LEE, SEONGJIN;YOO, CHANGWOO
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.22 no.1
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• pp.75-89
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• 2018
• This paper presents the numerical valuation of the arithmetic Asian option by using the operator-splitting method (OSM). Since there is no closed-form solution for the arithmetic Asian option, finding a good numerical algorithm to value the arithmetic Asian option is important. In this paper, we focus on a two-dimensional PDE. The OSM is famous for dealing with plural-dimensional PDE using finite difference discretization. We provide a detailed numerical algorithm and compare results with MCS method to show the performance of the method.

### Numerical simulation of non-isothermal flow in oil reservoirs using a two-equation model

• dos Santos Heringer, Juan Diego;de Souza Debossam, Joao Gabriel;de Souza, Grazione;Souto, Helio Pedro Amaral
• Coupled systems mechanics
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• v.8 no.2
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• pp.147-168
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• 2019
• This work aims to simulate three-dimensional heavy oil flow in a reservoir with heater-wells. Mass, momentum and energy balances, as well as correlations for rock and fluid properties, are used to obtain non-linear partial differential equations for the fluid pressure and temperature, and for the rock temperature. Heat transfer is simulated using a two-equation model that is more appropriate when fluid and rock have very different thermal properties, and we also perform comparisons between one- and two-equation models. The governing equations are discretized using the Finite Volume Method. For the numerical solution, we apply a linearization and an operator splitting. As a consequence, three algebraic subsystems of linearized equations are solved using the Conjugate Gradient Method. The results obtained show the suitability of the numerical method and the technical feasibility of heating the reservoir with static equipment.

### COMPARISON OF NUMERICAL SCHEMES ON MULTI-DIMENSIONAL BLACK-SCHOLES EQUATIONS

• Jo, Joonglee;Kim, Yongsik
• Bulletin of the Korean Mathematical Society
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• v.50 no.6
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• pp.2035-2051
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• 2013
• In this paper, we study numerical schemes for solving multi-dimensional option pricing problem. We compare the direct solving method and the Operator Splitting Method(OSM) by using finite difference approximations. By varying parameters of the Black-Scholes equations for the maximum on the call option problem, we observed that there is no significant difference between the two methods on the convergence criterion except a huge difference in computation cost. Therefore, the two methods are compatible in practice and one can improve the time efficiency by combining the OSM with parallel computation technique. We show numerical examples including the Equity-Linked Security(ELS) pricing based on either two assets or three assets by using the OSM with the Monte-Carlo Simulation as the benchmark.