• 대한토목학회논문집
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• 제13권3호
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• pp.129-138
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• 1993

본 연구는 Navier-Stokes 식의 모형방정식으로 이류 및 확산거동을 갖는 선형화된 Burgers 방정식과 비선형 형태의 Burgers 방정식을 선택하여, 이에 대한 유한차분법과 유한해석법의 수치해를 해석해와 비교하여 봄으로써, 유한해석법의 응용성에 대해 고찰한 것이다. 본 연구를 통하여 얻어진 성과를 요약하면 다음과 같다. Burgers 방정식 및 선형화된 Burgers 방정식의 정상상태의 해석해를 사용하여 두 수치기법에 따른 수치해를 비교해 본 결과, 해석해와의 근사정도를 동일 기준 하에서 살펴볼 때, 유한해석법이 유한차분법보다 우수한 것으로 나타났다. Burgers 방정식의 비정상상태의 해석해에 대한 정확성 또한 유한해석법이 보다 잘 일치하는 것으로 나타났다. 특히 유한해석법은 유한차분법의 사용시 격자 크기의 선택에 따라 해의 수렴과정에서 발생할 수 있는 위상오차에 기인한 진동현상이 전혀 발생하지 않는다는 것을 확인할 수 있었으며, 따라서 유한해석법은 수치기법상 위상오차로부터 자유로운 안정된 해석기법이라고 판단된다.

• 대한수학회보
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• 제45권1호
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• pp.191-205
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• 2008

In this paper, we construct the analytic solutions and numerical solutions for a two-dimensional Riemann problem for Burgers' equation. In order to construct the analytic solution, we use the characteristic analysis with the shock and rarefaction base points. We apply the composite scheme suggested by Liska and Wendroff to compute numerical solutions. The result is coincident with our analytic solution. This demonstrates that the composite scheme works pretty well for Burgers' equation despite of its simplicity.

• Journal of applied mathematics & informatics
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• 제11권1_2호
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• pp.81-108
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• 2003

We consider the Korteweg-de Vries-Burgers (KdVB) equation on the domain [0,11]. We derive a control law which guarantees $L^2$-global exponential stability, $H^3$-global asymptotic stability, and $H^3$-semiglobal exponential stability.

• 대한수학회논문집
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• 제35권2호
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• pp.685-695
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• 2020

In this paper, we used modified tanh-coth method, combined with Riccati equation and secant hyperbolic ansatz to construct abundantly many real and complex exact travelling wave solutions to KdV-Burgers (KdVB) equation with forcing term. The real part is the sum of the shock wave solution of a Burgers equation and the solitary wave solution of a KdV equation with forcing term, while the imaginary part is the product of a shock wave solution of Burgers with a solitary wave travelling solution of KdV equation. The method gives more solutions than the previous methods.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제13권4호
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• pp.293-305
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• 2009

In this article, we study a reduced-order modelling for distributed feedback control problem of the Burgers equations. Brief review of the centroidal Voronoi tessellation (CVT) are provided. A weighted (nonuniform density) CVT is introduced and low-order approximate solution and compensator-based control design of Burgers equation is discussed. Through weighted CVT (or CVT-nonuniform) method, obtained low-order basis is applied to low-order functional gains to design a low-order controller, and by using the low-order basis order of control modelling was reduced. Numerical experiments show that a solution of reduced-order controlled Burgers equation performs well in comparison with a solution of full order controlled Burgers equation.

• Korean Journal of Mathematics
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• 제24권4호
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• pp.637-645
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• 2016

In this paper, we study dynamical Bifurcation of the Burgers-Fisher equation. We show that the equation bifurcates an invariant set ${\mathcal{A}}_n({\beta})$ as the control parameter ${\beta}$ crosses over $n^2$ with $n{\in}{\mathbb{N}}$. It turns out that ${\mathcal{A}}_n({\beta})$ is homeomorphic to $S^1$, the unit circle.

• Journal of applied mathematics & informatics
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• 제26권3_4호
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• pp.573-581
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• 2008

In this paper, a finite difference method for a Cauchy problem of RLW-Burgers equation was considered. Although the equation is not energy conservation, we have given its the energy conservative finite difference scheme with condition. Convergence and stability of the difference solution were proved. Numerical results demonstrate that the method is efficient and reliable.

• Journal of applied mathematics & informatics
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• 제23권1_2호
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• pp.43-55
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• 2007

In this paper, an $H^1-Galerkin$ mixed finite element method is used to approximate the solution as well as the flux of Burgers' equation. Error estimates have been derived. The results of the numerical experiment show the efficacy of the mixed method and justifies the theoretical results obtained in the paper.

• 대한수학회지
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• 제54권1호
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• pp.193-213
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• 2017

We investigate an efficient approximation of solution to stochastic Burgers equation driven by an additive space-time noise. We discuss existence and uniqueness of a solution through the Orstein-Uhlenbeck (OU) process. To approximate the OU process, we introduce the Karhunen-$Lo{\grave{e}}ve$ expansion, and sparse grid stochastic collocation method. About spatial discretization of Burgers equation, two separate finite element approximations are presented: the conventional Galerkin method and Galerkin-conservation method. Numerical experiments are provided to demonstrate the efficacy of schemes mentioned above.

• Journal of applied mathematics & informatics
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• 제15권1_2호
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• pp.29-51
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• 2004

In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers' equation. This method is based upon a space-time variational form of Burgers' equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations. One of the contributions of the paper is to show how the optimal error estimates in $L^2(\Omega)$ are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.