• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제19권2호
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• pp.123-135
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• 2015

In this paper, a reduced-order modeling(ROM) of Burgers equations is studied based on pseudo-spectral collocation method. A ROM basis is obtained by the proper orthogonal decomposition(POD). Crank-Nicolson scheme is applied in time discretization and the pseudo-spectral element collocation method is adopted to solve linearlized equation based on the Newton method in spatial discretization. We deliver POD-based algorithm and present some numerical experiments to show the efficiency of our proposed method.

• Journal of Advanced Marine Engineering and Technology
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• 제22권5호
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• pp.670-679
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• 1998

The solution of hyperbolic equation with wave nature has sharp discontinuties in the medium at the wave front. Difficulties encounted in the numrtical solution of such problem in clude among oth-ers numerical oscillation and the representation of sharp discontinuities with good resolution at the wave front. In this work inviscid Burgers equation and modified heat conduction equation is intro-duced as hyperboic equation. These equations are caculated by numerical methods(explicit method MacCormack method Total Variation Diminishing(TVD) method) along various Courant numbers and numerical solutions are compared with the exact analytic solution. For inviscid Burgers equa-tion TVD method remains stable and produces high resolution at sharp wave front but for modified heat Conduction equation MacCormack method is recommmanded as numerical technique.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제22권4호
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• pp.289-294
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• 2018

In this work, a recently developed semi-analytic technique, so called the residual power series method, is generalized to process higher-dimensional linear and nonlinear partial differential equations. The solutions obtained takes a form of an infinite power series which can, in turn, be expressed in a closed exact form. The results reveal that the proposed generalization is very effective, convenient and simple. This is achieved by handling the (m+1)-dimensional Burgers equation.

• East Asian mathematical journal
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• 제34권5호
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• pp.661-681
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• 2018

In this paper, we discuss a reduced-order modeling for the Benjamin-Bona-Mahony-Burgers (BBMB) equation and its application to a distributed feedback control problem through the centroidal Voronoi tessellation (CVT). Spatial distcritization to the BBMB equation is based on the finite element method (FEM) using B-spline functions. To determine the basis elements for the approximating subspaces, we elucidate the CVT approaches to reduced-order bases with snapshots. For the purpose of comparison, a brief review of the proper orthogonal decomposition (POD) is provided and some numerical experiments implemented including full-order approximation, CVT based model, and POD based model. In the end, we apply CVT reduced-order modeling technique to a feedback control problem for the BBMB equation.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제18권3호
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• pp.269-277
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• 2014

A numerical scheme is proposed to control the BBMB (Benjamin-Bona-Mahony-Burgers) equation, and the scheme consists of three steps. Firstly, BBMB equation is converted to a finite set of nonlinear ordinary differential equations by the quadratic B-spline finite element method in spatial. Secondly, the controller is designed based on the linear quadratic regulator (LQR) theory; Finally, the system of the closed loop compensator obtained on the basis of the previous two steps is solved by the backward Euler method. The controlled numerical solutions are obtained for various values of parameters and different initial conditions. Numerical simulations show that the scheme is efficient and feasible.

• Journal of applied mathematics & informatics
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• 제29권3_4호
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• pp.813-829
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• 2011

In this article, we present a numerical scheme for solving singularly perturbed (i.e. highest -order derivative term multiplied by small parameter) Burgers-Huxley equation with appropriate initial and boundary conditions. Most of the traditional methods fail to capture the effect of layer behavior when small parameter tends to zero. The presence of perturbation parameter and nonlinearity in the problem leads to severe difficulties in the solution approximation. To overcome such difficulties the present numerical scheme is constructed. In construction of the numerical scheme, the first step is the dicretization of the time variable using forward difference formula with constant step length. Then, the resulting non linear singularly perturbed semidiscrete problem is linearized using quasi-linearization process. Finally, differential quadrature method is used for space discretization. The error estimate and convergence of the numerical scheme is discussed. A set of numerical experiment is carried out in support of the developed scheme.

• Kyungpook Mathematical Journal
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• 제34권1호
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• pp.21-41
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• 1994

• 한국수자원학회:학술대회논문집
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• 한국수자원학회 1988년도 제30회 수공학연구발표회논문초록집
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• pp.58-65
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• 1988

• Kyungpook Mathematical Journal
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• 제32권3_spc호
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• pp.587-606
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• 1992

• 융합보안논문지
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• 제5권3호
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• pp.93-97
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• 2005

MacCormack 방법은 hyperbolic 편미분 방정식의 근을 구하는데 많이 쓰이는 방법으로 그 정확도가 2차 오더가 된다. 하지만 이 방법으로 편미분방정식을 풀 경우 불연속인 점에서는 엔트로피를 만족하지 않는 경우가 있어 우리는 임의의 항을 첨가하여 근을 구해야한다. 이 임의의 항을 첨가하지 않고 직접 방정식으로부터 구하는 방법을 생각하는데 있어서 기존의 MacCormack 방법에 새 central scheme의 개념을 이용하면 전형적인 MacCormack 방법의 정확도와 장점을 보존할 수 있다. 이 새로운 방법을 이용하여 1D Burgers' 방정식과 1D Euler gas dynamic 방정식에 활용하여 그 결과를 살펴본다.