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Kim, Philsu,Kim, Sang Dong,Lee, Yong Hun Korean Mathematical Society 2016 대한수학회보 Vol.53 No.2
The bubble stabilization technique of Chebyshev-Legendre high-order element methods for one dimensional advection-diffusion equation is analyzed for the proposed scheme by Canuto and Puppo in [8]. We also analyze the finite element lower-order preconditioner for the proposed stabilized linear system. Further, the numerical results are provided to support the developed theories for the convergence and preconditioning.
An Enhanced Chebyshev Collocation Method Based on the Integration of Chebyshev Interpolation
Kim, Philsu Department of Mathematics 2017 Kyungpook mathematical journal Vol.57 No.2
In this paper, we develop an enhanced Chebyshev collocation method based on an integration scheme of the generalized Chebyshev interpolations for solving stiff initial value problems. Unlike the former error embedded Chebyshev collocation method (CCM), the enhanced scheme calculates the solution and its truncation error based on the interpolation of the derivative of the true solution and its integration. In terms of concrete convergence and stability analysis, the constructed algorithm turns out to have the $7^{th}$ convergence order and the A-stability without any loss of advantages for CCM. Throughout a numerical result, we assess the proposed method is numerically more efficient compared to existing methods.
Simple ECEM Algorithms Using Function Values Only
Kim, Philsu,Kim, Sang Dong,Lee, Eunjung Department of Mathematics 2013 Kyungpook mathematical journal Vol.53 No.4
In this paper, we improve the error corrected Euler method(ECEM) introduced in [11] by evaluating function values only at local nodes in each time interval. As a result, one can avoid computations of Jacobian matrices on each time interval so that the algorithms become simpler to implement in solving various class of time dependent differential equations numerically. The proposed ECEM formula resembles to the Runge-Kutta method in its representations but both methods have different characteristic properties.
A SURFACE RECONSTRUCTION METHOD FOR SCATTERED POINTS ON PARALLEL CROSS SECTIONS
PHILSU KIM 한국산업응용수학회 2005 Journal of the Korean Society for Industrial and A Vol.9 No.2
We consider a surface reconstruction problem from geometrical points(i.e., points given without any order) distributed on a series of smooth parallel cross sections in IR³. To solve the problem, we utilize the natural points ordering method in IR², described in [18], which is a method of reconstructing a curve from a set of sample points and is based on the concept of diffusion motions of a small object from one point to the other point. With only the information of the positions of these geometrical points, we construct an acceptable surface consisting of triangular facets using a heuristic algorithm to link a pair of parallel cross-sections constructed via the natural points ordering method. We show numerical simulations for the proposed algorithm with some sets of sample points.
Convergence on error correction methods for solving initial value problems
Kim, Sang Dong,Piao, Xiangfan,Kim, Do Hyung,Kim, Philsu Elsevier 2012 Journal of computational and applied mathematics Vol.236 No.17
<P><B>Abstract</B></P><P>Higher-order semi-explicit one-step error correction methods(ECM) for solving initial value problems are developed. ECM provides the excellent convergence O(<SUP>h2p+2</SUP>) one wants to get without any iteration processes required by most implicit type methods. This is possible if one constructs a local approximation having a residual error O(<SUP>hp</SUP>) on each time step. As a practical example, we construct a local quadratic approximation. Further, it is shown that special choices of parameters for the local quadratic polynomial lead to the known explicit second-order methods which can be improved into a semi-explicit type ECM of the order of accuracy 6. The stability function is also derived and numerical evidences are presented to support theoretical results with several stiff and non-stiff problems. It should be remarked that the ECM approach developed here does not yield explicit methods, but semi-implicit methods of the Rosenbrock type. Both ECM and Rosenbrock’s methods require to solve a few linear systems at each integration step, but the ECM approach involves 2p+2 evaluations of the Jacobian matrix per integration step whereas the Rosenbrock method demands one evaluation only. However, it is much easier to get high order methods by using the ECM approach.</P>
Kim, Philsu,Choi, Hyun Jung,Ahn, Soyoung Department of Mathematics 2007 Kyungpook mathematical journal Vol.47 No.4
The purpose of this article is to find a relation between the finite difference method and the boundary element method, and propose a new approach deriving a discrete approximation formula as like that of the finite difference method for harmonic functions. We develop a discrete approximation formula on a uniform grid based on the boundary integral formulations. We consider three different boundary integral formulations and derive one discrete approximation formula on the uniform grid for the harmonic function. We show that the proposed discrete approximation formula has the same computational molecules with that of the finite difference formula for the Laplace operator ${\nabla}^2$.
Error Control Strategy in Error Correction Methods
KIM, PHILSU,BU, SUNYOUNG Department of Mathematics 2015 Kyungpook mathematical journal Vol.55 No.2
In this paper, we present the error control techniques for the error correction methods (ECM) which is recently developed by P. Kim et al. [8, 9]. We formulate the local truncation error at each time and calculate the approximated solution using the solution and the formulated truncation error at previous time for achieving uniform error bound which enables a long time simulation. Numerical results show that the error controlled ECM provides a clue to have uniform error bound for well conditioned problems [1].
Philsu Kim,SANG DONG KIM,Yong Hun Lee 대한수학회 2016 대한수학회보 Vol.53 No.2
The bubble stabilization technique of Chebyshev-Legendre high-order element methods for one dimensional advection-diffusion equation is analyzed for the proposed scheme by Canuto and Puppo in \cite{CP}. We also analyze the finite element lower-order preconditioner for the proposed stabilized linear system. Further, the numerical results are provided to support the developed theories for the convergence and preconditioning.