14 results on '"Seongjai Kim"'
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2. A variable-θ method for parabolic problems of nonsmooth data
- Author
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Seongjai Kim and Philku Lee
- Subjects
Smoothness (probability theory) ,Speed wobble ,Numerical analysis ,Mathematical analysis ,Value (computer science) ,010103 numerical & computational mathematics ,01 natural sciences ,Stability (probability) ,010101 applied mathematics ,Set (abstract data type) ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,0101 mathematics ,Smoothing ,Mathematics ,Variable (mathematics) - Abstract
Parabolic initial–boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution. For those problems, specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution. This article investigates characteristics of the θ -method and introduces a variable- θ method as a synergistic combination of the Crank–Nicolson (CN) method ( θ = 1 ∕ 2 ) and the implicit method ( θ = 1 ). It suppresses spurious oscillations, by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness, and maintains as a similar accuracy as the CN method with smooth data. An effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations (the wobble set); the resulting variable- θ method is analyzed for its accuracy and stability. Various numerical examples are given to verify its effectiveness.
- Published
- 2020
3. Dual-Mesh Characteristics for Particle-Mesh Methods for the Simulation of Convection-Dominated Flows
- Author
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Chung-Ki Cho, Seongjai Kim, and Byungjoon Lee
- Subjects
Convection ,business.industry ,Applied Mathematics ,010103 numerical & computational mathematics ,Mechanics ,Computational fluid dynamics ,Solver ,Tracking (particle physics) ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Discontinuous Galerkin method ,Particle Mesh ,Particle ,0101 mathematics ,Diffusion (business) ,business ,Mathematics - Abstract
The particle-mesh method (PMM) is a powerful computational tool for the simulation of convection-dominated diffusion flows. The method introduces computational particles each of which is given a finite size and represents a large number of physical particles with the same properties. The convection part of the flow can be solved by moving the computational particles along the characteristics, while the diffusion part is carried out by utilizing a heat solver on a regular mesh. However, the method in practical applications shows the so-called ringing instability, an amplitude fluctuation in the computed solution. In this article, we suggest a new numerical technique of particle movement, called the dual-mesh characteristics (DMC) of which the second mesh is formed by tracking back the cells along the characteristics. The particle movement is carried out by interpreting the particle positions (in the previous time level) in terms of the multilinear coordinates of the second mesh. Strategies for the average ...
- Published
- 2018
4. An Essentially Non-oscillatory Crank–Nicolson Procedure for the Simulation of Convection-Dominated Flows
- Author
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Byungjoon Lee, Seongjai Kim, and Myungjoo Kang
- Subjects
Convection ,Numerical Analysis ,Applied Mathematics ,Courant–Friedrichs–Lewy condition ,Mathematical analysis ,General Engineering ,Upwind scheme ,010103 numerical & computational mathematics ,Dissipation ,01 natural sciences ,Stability (probability) ,Theoretical Computer Science ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Crank–Nicolson method ,0101 mathematics ,Diffusion (business) ,Software ,Mathematics ,Convection dominated - Abstract
The Crank–Nicolson (CN) time-stepping procedure incorporating the second-order central spatial scheme is unconditionally stable and strictly non-dissipative for linear convection flows; however, its numerical solution in practice can be oscillatory for nonsmooth solutions. This article studies variants of the CN method for the simulation of linear convection-dominated diffusion flows, in which the explicit convection part is approximated by an upwind scheme, to effectively suppress nonphysical oscillations. The second-order essentially non-oscillatory scheme incorporated in the CN procedure (ENO-CN) has been found effective for a non-oscillatory numerical solution of minimum numerical dissipation. A stability analysis is provided for ENO-CN, which turns out to be unconditionally stable for problems of nonzero diffusion. However, for purely convective flows, it is stable only when the CFL condition is satisfied. Numerical results are presented to demonstrate its stability and accuracy.
- Published
- 2016
5. High-order schemes for acoustic waveform simulation
- Author
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Hyeona Lim and Seongjai Kim
- Subjects
Computational Mathematics ,Numerical Analysis ,Applied Mathematics ,Numerical analysis ,Explicit and implicit methods ,Stability (learning theory) ,Waveform ,Acoustic wave equation ,Acoustic wave ,Wave equation ,Algorithm ,Numerical stability ,Mathematics - Abstract
This article introduces a new fourth-order implicit time-stepping scheme for the numerical solution of the acoustic wave equation, as a variant of the conventional modified equation method. For an efficient simulation, the scheme incorporates a locally one-dimensional (LOD) procedure having the splitting error of O(@Dt^4). Its stability and accuracy are compared with those of the standard explicit fourth-order scheme. It has been observed from various experiments for 2D problems that (a) the computational cost of the implicit LOD algorithm is only about 40% higher than that of the explicit method, for the problems of the same size, (b) the implicit LOD method produces less dispersive solutions in heterogeneous media, and (c) its numerical stability and accuracy match well those of the explicit method.
- Published
- 2007
6. Numerical methods for viscous and nonviscous wave equations
- Author
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Seongjai Kim, Hyeona Lim, and Jim Douglas
- Subjects
Numerical Analysis ,Wave propagation ,Applied Mathematics ,Numerical analysis ,Diagonal ,Perturbation (astronomy) ,Geometry ,Acoustic wave ,Wave equation ,Computational Mathematics ,Heat transfer ,Applied mathematics ,Microscale chemistry ,Mathematics - Abstract
This article is concerned with accurate and efficient numerical methods for solving viscous and nonviscous wave equations. A three-level second-order implicit algorithm is considered without introducing auxiliary variables. As a perturbation of the algorithm, a locally one-dimensional (LOD) procedure which has a splitting error not larger than the truncation error is suggested to solve problems of diagonal diffusion tensors in cubic domains efficiently. Both the three-level algorithm and its LOD procedure are proved to be unconditionally stable. An error analysis is provided for the numerical solution of viscous waves. Numerical results are presented to show the accuracy and efficiency of the new algorithms for the propagation of acoustic waves and of microscale heat transfer.
- Published
- 2007
7. Multigrid Simulation for High-Frequency Solutions of the Helmholtz Problem in Heterogeneous Media
- Author
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Seongjai Kim and Soohyun Kim
- Subjects
Computational Mathematics ,Mathematical optimization ,Partial differential equation ,Multigrid method ,Rate of convergence ,Helmholtz equation ,Iterative method ,Applied Mathematics ,Applied mathematics ,Domain decomposition methods ,Decomposition method (constraint satisfaction) ,Grid ,Mathematics - Abstract
The Helmholtz problem is hard to solve in heterogeneous media, in particular, when the wave number is real and large. The problem is neither coercive nor Hermitian symmetric. This article concerns the V-cycle multigrid (MG) method for high-frequency solutions of the Helmholtz problem. Since we need to choose at least 10--12 grid points per wavelength for stability, the coarse grid problem is still large. To solve the coarse grid problem efficiently, a nonoverlapping domain decomposition method is adopted without introducing another coarser subspace correction. Various numerical experiments have shown that the convergence rate of the resulting MG method is independent on the grid size and the wave number, provided that the coarse grid problem is fine enough for the solution to capture characteristics of the physical problem.
- Published
- 2002
8. An $\cal O(N)$ Level Set Method for Eikonal Equations
- Author
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Seongjai Kim
- Subjects
Computational Mathematics ,Level set (data structures) ,Partial differential equation ,Level set method ,Eikonal equation ,Applied Mathematics ,Finite difference method ,Applied mathematics ,Geometry ,Classification of discontinuities ,Grid ,Fast marching method ,Mathematics - Abstract
A propagating interface can develop corners and discontinuities as it advances. Level set algorithms have been extensively applied for the problems in which the solution has advancing fronts. One of the most popular level set algorithms is the so-called {fast marching method} (FMM), which requires total $\cal O(N\log_2N)$ operations, where N is the number of grid points. The article is concerned with the development of an $\cal O(N)$ level set algorithm called the group marching method (GMM). The new method is based on the narrow band approach as in the FMM. However, it is incorporating a correction-by-iteration strategy to advance a group of grid points at a time, rather than sorting the solution in the narrow band to march forward a single grid point. After selecting a group of grid points appropriately, the GMM advances the group in two iterations for the cost of slightly larger than one iteration. Numerical results are presented to show the efficiency of the method, applied to the eikonal equation in two and three dimensions.
- Published
- 2001
9. Wavefronts of linear elastic waves: local convexity and modeling
- Author
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Seongjai Kim
- Subjects
Wavefront ,Eikonal equation ,Applied Mathematics ,Isotropy ,Linear elasticity ,Mathematical analysis ,Finite difference ,General Physics and Astronomy ,Tangent ,Classification of discontinuities ,Convexity ,Computational Mathematics ,Modeling and Simulation ,Mathematics - Abstract
Seismic techniques incorporating high frequency asymptotic representation of the 3D elastic Green’s function require efficient solution methods for the computation of traveltimes. For finite difference eikonal solvers, upwind differences are requisite to sharply resolve discontinuities in the traveltime derivatives. In anisotropic media, the direction of energy propagation is not in general tangent to the wavefront normal, while finite difference eikonal solvers compute the solution based on the traveltime gradients and wavefront normal. Local convexity of the wavefronts in transverse isotropic (TI) media is proved to show that wavefront normal determines the upwind direction of the energy propagation. The eikonal equations for the traveltimes in TI media of a generally inclined symmetry axis (ITI) are derived in a way that the eikonal solvers fit conveniently. A stable, second-order, shock-capturing, upwind finite difference scheme is suggested for solving ITI eikonal equations in regular grids in 3D. Numerical experiments are presented to demonstrate the efficiency of the algorithm.
- Published
- 2000
10. On the use of rational iterations and domain decomposition methods for the Helmholtz problem
- Author
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Seongjai Kim
- Subjects
Computational Mathematics ,Mathematical optimization ,Helmholtz equation ,Iterative method ,Applied Mathematics ,Numerical analysis ,Parallel algorithm ,Boundary (topology) ,Applied mathematics ,Domain decomposition methods ,Boundary value problem ,Relaxation (approximation) ,Mathematics - Abstract
An iterative algorithm for the numerical solution of the Helmholtz problem is considered. It is difficult to solve the problem numerically, in particular, when the imaginary part of the wave number is zero or small. We develop a parallel iterative algorithm based on a rational iteration and a nonoverlapping domain decomposition method for such a non-Hermitian, non-coercive problem. Algorithm parameters (artificial damping and relaxation) are introduced to accelerate the convergence speed of the iteration. Convergence analysis and effective strategies for finding efficient algorithm parameters are presented. Numerical results carried out on an nCUBE2 are given to show the efficiency of the algorithm. To reduce the boundary reflection, we employ a hybrid absorbing boundary condition (ABC) which combines the first-order ABC and the physical $Q$ ABC. Computational results comparing the hybrid ABC with non-hybrid ones are presented.
- Published
- 1998
11. Domain decomposition iterative procedures for solving scalar waves in the frequency domain
- Author
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Seongjai Kim
- Subjects
Computational Mathematics ,Rate of convergence ,Fictitious domain method ,Iterative method ,Applied Mathematics ,Frequency domain ,Mathematical analysis ,Relaxation (iterative method) ,Domain decomposition methods ,Mortar methods ,Domain (software engineering) ,Mathematics - Abstract
The propagation of dispersive waves can be modeled relevantly in the frequency domain. A wave problem in the frequency domain is difficult to solve numerically. In addition to having a complex–valued solution, the problem is neither Hermitian symmetric nor coercive in a wide range of applications in Geophysics or Quantum–Mechanics. In this paper, we consider a parallel domain decomposition iterative procedure for solving the problem by finite differences or conforming finite element methods. The analysis includes the decomposition of the domain into either the individual elements or larger subdomains (\(blocks\) of finite elements). To accelerate the speed of convergence, we introduce relaxation parameters on the subdomain interfaces and an artificial damping iteration. The convergence rate of the resulting algorithm turns out to be independent on the mesh size and the wave number. Numerical results carried out on an nCUBE2 parallel computer are presented to show the effectiveness of the method.
- Published
- 1998
12. Parallel multidomain iterative algorithms for the Helmholtz wave equation
- Author
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Seongjai Kim
- Subjects
Numerical Analysis ,Partial differential equation ,Helmholtz equation ,Iterative method ,Applied Mathematics ,Parallel algorithm ,Domain decomposition methods ,Wave equation ,Computational Mathematics ,symbols.namesake ,Parallel processing (DSP implementation) ,Helmholtz free energy ,symbols ,Algorithm ,Mathematics - Abstract
In this paper, we consider parallel iterative algorithms for solving the Helmholtz wave equation employing nonoverlapping domain decomposition techniques. A modified Robin interface condition incorporated with an iteration parameter is used to communicate the data near the interfaces. An automatic and non-expensive strategy for finding efficient iteration parameters is discussed in detail. Numerical results carried out on an nCUBE2 are given to demonstrate the effectiveness of the method.
- Published
- 1995
13. A parallelizable iterative procedure for the Helmholtz problem
- Author
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Seongjai Kim
- Subjects
Computational Mathematics ,Numerical Analysis ,Parallelizable manifold ,Helmholtz problem ,Helmholtz equation ,Iterative method ,Applied Mathematics ,Mathematical analysis ,Finite difference ,Finite difference method ,Domain decomposition methods ,Boundary value problem ,Mathematics - Abstract
A parallelizable iterative procedure based on domain decomposition techniques is defined and analyzed for finite difference approximate solutions for the Helmholtz problem. An automatic efficient strategy for choosing the algorithm parameter is presented. Numerical results are reported.
- Published
- 1994
14. Artificial damping techniques for scalar waves in the frequency domain
- Author
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M. Lee and Seongjai Kim
- Subjects
Partial differential equation ,Wave propagation ,Helmholtz equation ,Numerical analysis ,Mathematical analysis ,MathematicsofComputing_NUMERICALANALYSIS ,Domain decomposition methods ,Robin boundary condition ,Wave equation ,Computational Mathematics ,Computational Theory and Mathematics ,Modelling and Simulation ,Modeling and Simulation ,Frequency domain ,Convergence (routing) ,Domain decomposition method ,Artificial damping ,Scalar field ,Mathematics - Abstract
An artificial damping algorithm for solving the Helmholtz problem is considered. When the imaginary part of the wave number is small, the problem is known to be difficult to solve. In this paper, we propose an efficient artificial damping algorithm which can be viewed as a rational iteration. Each damped problem is solved incompletely by a nonoverlapping domain decomposition method. Convergence arguments are presented. Numerical results run on an nCUBE2 are presented to demonstrate the effectiveness of the method.
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