Relevant bibliographies by topics / Shishkin-type mesh

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  2. Dissertations / Theses

Academic literature on the topic 'Shishkin-type mesh'

Author: Grafiati
Published: 4 June 2021
Last updated: 14 February 2022

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Journal articles on the topic "Shishkin-type mesh"

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Nhan, Thái Anh, and Relja Vulanović. "A note on a generalized Shishkin-type mesh." Novi Sad Journal of Mathematics 48, no. 2 (May 4, 2018): 141–50. http://dx.doi.org/10.30755/nsjom.07880.

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Cakir, Musa. "A NUMERICAL STUDY ON THE DIFFERENCE SOLUTION OF SINGULARLY PERTURBED SEMILINEAR PROBLEM WITH INTEGRAL BOUNDARY CONDITION." Mathematical Modelling and Analysis 21, no. 5 (September 20, 2016): 644–58. http://dx.doi.org/10.3846/13926292.2016.1201702.

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The present study is concerned with the numerical solution, using finite difference method on a piecewise uniform mesh (Shishkin type mesh) for a singularly perturbed semilinear boundary value problem with integral boundary condition. First we discuss the nature of the continuous solution of singularly perturbed differential problem before presenting method for its numerical solution. The numerical method is constructed on piecewise uniform Shishkin type mesh. We show that the method is first-order convergent in the discrete maximum norm, independently of singular perturbation parameter except for a logarithmic factor. We give effective iterative algorithm for solving the nonlinear difference problem. Numerical results which support the given estimates are presented.
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Govindarao, Lolugu, and Jugal Mohapatra. "A second order numerical method for singularly perturbed delay parabolic partial differential equation." Engineering Computations 36, no. 2 (March 11, 2019): 420–44. http://dx.doi.org/10.1108/ec-08-2018-0337.

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Purpose The purpose of this paper is to provide a robust second-order numerical scheme for singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Design/methodology/approach For the parabolic convection-diffusion initial boundary value problem, the authors solve the problem numerically by discretizing the domain in the spatial direction using the Shishkin-type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh) and in temporal direction using the uniform mesh. The time derivative is discretized by the implicit-trapezoidal scheme, and the spatial derivatives are discretized by the hybrid scheme, which is a combination of the midpoint upwind scheme and central difference scheme. Findings The authors find a parameter-uniform convergent scheme which is of second-order accurate globally with respect to space and time for the singularly perturbed delay parabolic convection–diffusion initial boundary value problem. Also, the Thomas algorithm is used which takes much less computational time. Originality/value A singularly perturbed delay parabolic convection–diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a hybrid scheme. The method is parameter-uniform convergent and is of second order accurate globally with respect to space and time. Numerical results are carried out to verify the theoretical estimates.
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Zhu, Huiqing, and Zhimin Zhang. "Pointwise Error Estimates for the LDG Method Applied to 1-d Singularly Perturbed Reaction-Diffusion Problems." Computational Methods in Applied Mathematics 13, no. 1 (January 1, 2013): 79–94. http://dx.doi.org/10.1515/cmam-2012-0004.

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Abstract. The local discontinuous Galerkin method (LDG) is considered for solving one-dimensional singularly perturbed two-point boundary value problems of reaction-diffusion type. Pointwise error estimates for the LDG approximation to the solution and its derivative are established on a Shishkin-type mesh. Numerical experiments are presented. Moreover, a superconvergence of order of the numerical traces is observed numerically.
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Mahendran, R., and V. Subburayan. "Fitted Finite Difference Method for Third Order Singularly Perturbed Delay Differential Equations of Convection Diffusion Type." International Journal of Computational Methods 16, no. 05 (May 28, 2019): 1840007. http://dx.doi.org/10.1142/s0219876218400078.

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In this paper, a fitted finite difference method on Shishkin mesh is suggested to solve a class of third order singularly perturbed boundary value problems for ordinary delay differential equations of convection-diffusion type. Numerical solution converges uniformly to the exact solution. The order of convergence of the numerical method is almost first order. Numerical results are provided to illustrate the theoretical results.
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Cimen, Erkan. "NUMERICAL SOLUTION OF A BOUNDARY VALUE PROBLEM INCLUDING BOTH DELAY AND BOUNDARY LAYER." Mathematical Modelling and Analysis 23, no. 4 (October 9, 2018): 568–81. http://dx.doi.org/10.3846/mma.2018.034.

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Difference method on a piecewise uniform mesh of Shishkin type, for a singularly perturbed boundary-value problem for a nonlinear second order delay differential equation is analyzed. Also, the method is proved that it gives essentially first order parameter-uniform convergence in the discrete maximum norm. Furthermore, numerical results are presented in support of the theory.
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KUMAR, VINOD, R. K. BAWA, and A. K. LAL. "A ROBUST COMPUTATIONAL TECHNIQUE FOR A SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS." International Journal of Computational Methods 10, no. 05 (May 2013): 1350027. http://dx.doi.org/10.1142/s0219876213500278.

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In this paper, a singularly perturbed system of convection-diffusion boundary value problem (BVP) is examined. To solve such type of problem, a modified initial value technique (MIVT) is proposed on an appropriate piecewise uniform Shishkin mesh. The MIVT is shown to be uniformly convergent with respect to the perturbation parameter. Numerical results are presented which are in agreement with the theoretical results.
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TAMILSELVAN, A., and N. RAMANUJAM. "AN ALMOST-SECOND-ORDER METHOD FOR A SYSTEM OF SINGULARLY PERTURBED CONVECTION–DIFFUSION EQUATIONS WITH NONSMOOTH CONVECTION COEFFICIENTS AND SOURCE TERMS." International Journal of Computational Methods 07, no. 02 (June 2010): 261–77. http://dx.doi.org/10.1142/s0219876210002167.

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In this paper, a weakly coupled system of two singularly perturbed convection–diffusion equations with discontinuous convection coefficients and source terms with Dirichlet type boundary conditions is considered. A hybrid finite difference scheme on a Shishkin mesh generating almost-second-order convergence in the maximum norm is constructed for solving this problem. To illustrate the theoretical results, numerical experiments are performed.
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9

Kadalbajoo, Mohan K., and Ashish Awasthi. "Parameter free hybrid numerical method for solving modified Burgers’ equations on a nonuniform mesh." Asian-European Journal of Mathematics 10, no. 02 (July 21, 2016): 1750029. http://dx.doi.org/10.1142/s1793557117500292.

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In this paper, the modified Burgers’ equation is considered. These kinds of problems come from the field of sonic boom and explosions theories. At big Reynolds’ number there is a boundary layer in the right side of the domain. From numerical point of view, the major difficulty in dealing with this type of problem is that the smooth initial data can give rise to solution varying regions i.e. boundary layer regions. To tackle this situation, we propose a numerical method on nonuniform mesh of Shishkin type, which works well at high as well as low Reynolds number. The proposed method comprises of Euler implicit scheme and hybrid scheme in time and space direction, respectively. First, we discretize the continuous problem in temporal direction by Euler implicit method, which yields a set of ode’s at each time level. The resulting set of differential equations are approximated by a hybrid scheme on Shishkin mesh i.e. upwind in regular region (nonboundary layer region) and central difference in boundary layer regions. The convergence of proposed method has been shown parameter uniform. Some numerical experiments have been carried out to corroborate the theoretical results.
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Çakır, Musa, and Gabil M. Amiraliyev. "A Numerical Method for a Singularly Perturbed Three-Point Boundary Value Problem." Journal of Applied Mathematics 2010 (2010): 1–17. http://dx.doi.org/10.1155/2010/495184.

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The purpose of this paper is to present a uniform finite difference method for numerical solution of nonlinear singularly perturbed convection-diffusion problem with nonlocal and third type boundary conditions. The numerical method is constructed on piecewise uniform Shishkin type mesh. The method is shown to be convergent, uniformly in the diffusion parameterε, of first order in the discrete maximum norm. Some numerical experiments illustrate in practice the result of convergence proved theoretically.
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Dissertations / Theses on the topic "Shishkin-type mesh"

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Fröhner, Anja. "Defektkorrekturverfahren für singulär gestörte Randwertaufgaben." Doctoral thesis, Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2002. http://nbn-resolving.de/urn:nbn:de:swb:14-1043050731546-74245.

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Wir untersuchen ein Defektkorrekturverfahren, das ein einfaches Upwind-Differenzenverfahren erster Ordnung mit einem zentralen Differenzenverfahren kombiniert, für ein- und zweidimensionale singulär gestörte Konvektions-Diffusions-Probleme auf einer Klasse von Shishkin-Typ-Gittern. Im eindimensionalen Fall wird nachgewiesen, dass das Verfahren von (fast) zweiter Ordnung, gleichmäßig bezüglich des Diffusionsparameters $\epsilon$ konvergiert. Zur Konvergenzanalyse für das zweidimensionale Modellproblem werden verschiedene Techniken diskutiert. In einem Spezialfall kann auf einem stückweise uniformen Shishkin-Gitter die $\epsilon$-gleichmäßige Konvergenz des Verfahrens von fast zweiter Ordnung gezeigt werden. Ferner sind die bisher bekannten Stabilitätsaussagen und ihre Verwendung zur Konvergenzanalysis der betrachteten Differenzenverfahren sowie Methoden zur Analyse von Defektkorrekturverfahren zusammengestellt. Einige Bemerkungen zu Defektkorrekturverfahren und Finite-Elemente-Methoden schließen die Arbeit ab. Numerische Experimente untermauern die theoretischen Resultate
We consider a defect correction method that combines a first-order upwinded difference scheme with a second-order central difference scheme for model singularly perturbed convection-diffusion problems in one and two dimensions on a class of Shishkin-Type meshes. In one dimension, the method is shown to be convergent uniformly in the diffusion parameter $\epsilon$ of second order in the discrete maximum norm. To analyze the two-dimensional case, we discuss several proof techniques for defect correction methods. For a special problem with constant coefficients on a piecewise uniform Shishkin-mesh we can show the second order convergence of the considered scheme, uniformly with respect to the diffusion parameter. Moreover the known stability properties and their impact on the convergence analysis of the considered differnce schemes are compiled. Some remarks on defect correction and finite elements conclude the theses. Numerical experiments support our theoretical results
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