• 대한수학회지
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• 제51권4호
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• pp.679-702
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• 2014

Investigation of the numerical solution of singularly perturbed turning point problems dates back to late 1970s. However, due to the presence of layers, not many high order schemes could be developed to solve such problems. On the other hand, one could think of applying the convergence acceleration technique to improve the performance of existing numerical methods. However, that itself posed some challenges. To this end, we design and analyze a novel fitted operator finite difference method (FOFDM) to solve this type of problems. Then we develop a fitted mesh finite difference method (FMFDM). Our detailed convergence analysis shows that this FMFDM is robust with respect to the singular perturbation parameter. Then we investigate the effect of Richardson extrapolation on both of these methods. We observe that, the accuracy is improved in both cases whereas the rate of convergence depends on the particular scheme being used.

• Journal of applied mathematics & informatics
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• 제22권1_2호
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• pp.49-65
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• 2006

A robust numerical method for a singularly perturbed second-order ordinary differential equation having two parameters with a discontinuous source term is presented in this article. Theoretical bounds are derived for the derivatives of the solution and its smooth and singular components. An appropriate piecewise uniform mesh is constructed, and classical upwind finite difference schemes are used on this mesh to obtain the discrete system of equations. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are provided to illustrate the convergence of the numerical approximations.

• Journal of applied mathematics & informatics
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• 제27권3_4호
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• pp.517-529
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• 2009

In this paper, a singularly perturbed reaction-convection-diffusion problem with two parameters is considered. A parameter-uniform error bound for the numerical derivative is derived. The numerical method considered here is a standard finite difference scheme on piecewise-uniform Shishkin mesh, which is fitted to both boundary and initial layers. Numerical results are provided to illustrate the theoretical results.

• Journal of applied mathematics & informatics
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• 제38권1_2호
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• pp.113-132
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• 2020

In this paper, a class of linear second order singularly perturbed delay differential turning point problems containing a small delay (or negative shift) on the reaction term and when the solution of the problem exhibits twin boundary layers are examined. A hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the problem. We proved that the method is almost second order ε-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results.

• Journal of Advanced Marine Engineering and Technology
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• 제33권6호
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• pp.873-879
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• 2009

The steady breakers at an infant stage are investigated through the numerical simulation. The appearing condition and characteristics of the sub-breaking waves are reviewed by analysing bow waves. The instability analysis is possibly done through the relationship between the free-surface curvature and circumferential force, which is obtained from the momentum equations. Navier-Stokes equations are solved by a finite difference method where the body-fitted coordinate system, the wall function and the advanced mesh system are invoked. The numerical result shows that the gradient of M/$U_s$ is greatly influenced by the Froude number and the decrease of M/$U_s$ indicates that the flows are unstable. Additionally flows with plunging or spilling are simulated successfully, but the application of breakers to the severely broken wave still remains to be settled in the future.