• Journal of applied mathematics & informatics
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• v.32 no.3_4
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• pp.511-527
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• 2014

This paper provides a new application similar to the Local Defect Correction (LDC) technique to solve Poisson problem -u"(x) = f(x) with Dirichlet boundary conditions. The exact solution is supposed to have high activity in some region of the domain. LDC is combined with a fourth order compact scheme which is recently developed in Abbas (Num. Meth. Partial differential equations, 2013). Numerical tests illustrate the interest of this application.

• Journal of the Korean Mathematical Society
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• v.59 no.5
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• pp.891-909
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• 2022

The main objective of this paper is to develop a concrete inverse formula of the system induced by the fourth-order finite difference method for two-point boundary value problems with Robin boundary conditions. This inverse formula facilitates to make a fast algorithm for solving the problems. Our numerical results show the efficiency and accuracy of the proposed method, which is implemented by the Thomas algorithm.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.297-310
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• 2018

In this paper, a nonlinear high-order difference scheme is proposed to solve the Extended-Fisher-Kolmogorov equation. The existence, uniqueness of difference solution and priori estimates are obtained. Furthermore, the convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete $L^{\infty}-norm$. Some numerical examples are given in order to validate the theoretical results.

• The Journal of the Acoustical Society of Korea
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• v.25 no.1E
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• pp.32-35
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• 2006

The screech tone of underexpanded jet is numerically calculated without any specific modeling for the screech tone itself. Fourth-order optimized compact scheme and fourth-order Runge-Kutta method are used to solve the 2D axisymmetric Euler equation. Adaptive nonlinear artificial dissipation model and generalized characteristic boundary condition are also used. The screech tone, generated by a closed loop between instability waves and quasi-periodic shock cells at the near field, is reasonably analyzed with present numerical methods for the underexpanded jet having Mach number 1.13. First of all, the centerline mean pressure distribution is calculated and compared with experimental and other numerical results. The instantaneous density contour plot shows Mach waves due to mixing layer convecting supersonically, which propagate downstream. The pressure signal and its Fourier transform at upstream and downstream shows the directivity pattern of screech tone very clearly. Most of all, we can simulate the axisymmetric mode change of screech tone very precisely with present method. It can be concluded that the basic phenomenon of screech tone including the frequency can be calculated by using high-order and high-resolution schemes without any specific numerical modeling for screech tone feedback loop.

• Journal of applied mathematics & informatics
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• v.34 no.5_6
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• pp.495-508
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• 2016

A singularly perturbed reaction-diffusion fourth-order ordinary differential equation(ODE) with discontinuous source term is considered. Due to the discontinuity, interior layers also exist. The considered problem is converted into a system of weakly coupled system of two second-order ODEs, one without parameter and another with parameter ε multiplying highest derivatives and suitable boundary conditions. In this paper a computational method for solving this system is presented. A zero-order asymptotic approximation expansion is applied in the second equation. Then, the resulting equation is solved by the numerical method which is constructed. This involves non-overlapping Schwarz method using Shishkin mesh. The computation shows quick convergence and results presented numerically support the theoretical results.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.11-27
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• 2003

A fourth order in time and second order in space scheme using a finite-difference method is developed for the non-linear Boussinesq equation. For the solution of the resulting non-linear system a predictor-corrector pair is proposed. The method is analyzed for local truncation error and stability. The results of a number of numerical experiments for both the single and the double-soliton waves are given.

• Journal of Advanced Marine Engineering and Technology
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• v.23 no.6
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• pp.794-805
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• 1999

Pneumatic cushioning cylinders are commonly employed for vibration and shock control. A mathematical simulation model of a double acting pneumatic cushioning cylinder designed to absorb shock loads is presented which is based on the following assumptions; ideal equation of state isentropic flow through a port conservation of mass polytropic thermodynamics single degree of freedom piston dynamics and energy equivalent linear damping. These differential equation can be solved through numerical integration using the fourth order Runge-Kutta method. An experimental study was conducted to validate the results obtained by the numerical integra-tion technique. Simulated results show good agreement with experimental data. The computer simulation model presented here has been extremely useful not only in understanding the has been extremely useful not only in understanding the basic cushioning but also in evaluating different designs.

• Journal of applied mathematics & informatics
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• v.8 no.1
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• pp.165-180
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• 2001

This paper presents a new difference scheme for numerical solution of stiff system of ODE’s. The present study is mainly motivated to develop an absolutely stable numerical method with a high order of approximation. In this work a double implicit A-stable difference scheme with the sixth order of approximation is suggested. Another purpose of this study is to introduce automatic choice of the integration step size of the difference scheme which is derived from the proposed scheme and the one step scheme of the fourth order of approximation. The algorithm was tested by means of solving the Kreiss problem and a chemical kinetics problem. The behavior of the gas explosive mixture (H₂+ O₂) in a closed space with a mobile piston is considered in test problem 2. It is our conclusion that a hydrogen-operated engine will permit to decrease the emitted levels of hazardous atmospheric pollutants.

• Publications of The Korean Astronomical Society
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• v.13 no.1 s.14
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• pp.167-180
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• 1998

We have carried out high precision orbit calculation, by using various numerical techniques with accuracy of higher than fourth order, in order for exact prediction on position and velocity of celestial bodies and artificial satellites. General second order ordinary differential equation has been solved numerically to test the performance for each of numerical methods. We have compared computed values with exact solution obtained by using universal variables for two body problem and discussed overall results of numerical methods used in our calculation. As a result, it is found that high order difference table method called as Gauss-Jackson method is best one with easiness and efficiency in the increase of accuracy by number of initial values.

• Korean Journal of Mathematics
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• v.13 no.2
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• pp.161-176
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• 2005

We consider a model of population dynamics whose mortality function is unbounded. We note that the regularity of the solution depends on the growth rate of the mortality near the maximum age. We propose Gauss-Legendre methods along the characteristics to approximate the solution when the solution is smooth enough. It is proven that the scheme is convergent at fourth-order rate in the maximum norm. We also propose discontinuous Galerkin finite element methods to approximate the solution which is not smooth enough. The stability of the method is discussed. Several numerical examples are presented.