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Journal articles on the topic 'Shishkin-type mesh'

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Published: 4 June 2021
Last updated: 14 February 2022

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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10

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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11

Kumar, Vinod, Rajesh K. Bawa, and Arvind K. Lal. "A robust computational technique for a system of singularly perturbed reaction–diffusion equations." International Journal of Applied Mathematics and Computer Science 24, no. 2 (June 26, 2014): 387–95. http://dx.doi.org/10.2478/amcs-2014-0029.

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Abstract In this paper, a singularly perturbed system of reaction–diffusion Boundary Value Problems (BVPs) is examined. To solve such a type of problems, a Modified Initial Value Technique (MIVT) is proposed on an appropriate piecewise uniform Shishkin mesh. The MIVT is shown to be of second order convergent (up to a logarithmic factor). Numerical results are presented which are in agreement with the theoretical results.
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12

Gelu, Fasika Wondimu, and Gemechis File Duressa. "A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem." Abstract and Applied Analysis 2021 (March 12, 2021): 1–11. http://dx.doi.org/10.1155/2021/8835595.

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In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be ε -uniformly convergent of O Δ t + N − 2 ln 2 N , where Δ t and N denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter λ . Some numerical results are carried out to support the theory.
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13

Priyadharshini, Rajarammohanroy Mythili, and Narashimhan Ramanujam. "UNIFORMLY-CONVERGENT NUMERICAL METHODS FOR A SYSTEM OF COUPLED SINGULARLY PERTURBED CONVECTION–DIFFUSION EQUATIONS WITH MIXED TYPE BOUNDARY CONDITIONS." Mathematical Modelling and Analysis 18, no. 5 (December 1, 2013): 557–98. http://dx.doi.org/10.3846/13926292.2013.851629.

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In this paper, two hybrid difference schemes on the Shishkin mesh are constructed for solving a weakly coupled system of two singularly perturbed convection - diffusion second order ordinary differential equations subject to the mixed type boundary conditions. We prove that the method has almost second order convergence in the supremum norm independent of the diffusion parameter. Error bounds for the numerical solution and also the numerical derivative are established. Numerical results are provided to illustrate the theoretical results.
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14

Podila, ramod Chakravarthy, Trun Gupta, and Nageshwar Rao. "A NUMERICAL SCHEME FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS OF CONVECTION-DIFFUSION TYPE ON AN ADAPTIVE GRID." Mathematical Modelling and Analysis 23, no. 4 (October 9, 2018): 686–98. http://dx.doi.org/10.3846/mma.2018.041.

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In this paper, an adaptive mesh strategy is presented for solving singularly perturbed delay differential equation of convection-diffusion type using second order central finite difference scheme. Layer adaptive meshes are generated via an entropy production operator. The details of the location and width of the layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.
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15

Nhan, Thái Anh, and Relja Vulanović. "Preconditioning and Uniform Convergence for Convection-Diffusion Problems Discretized on Shishkin-Type Meshes." Advances in Numerical Analysis 2016 (February 28, 2016): 1–11. http://dx.doi.org/10.1155/2016/2161279.

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A one-dimensional linear convection-diffusion problem with a perturbation parameter ɛ multiplying the highest derivative is considered. The problem is solved numerically by using the standard upwind scheme on special layer-adapted meshes. It is proved that the numerical solution is ɛ-uniform accurate in the maximum norm. This is done by a new proof technique in which the discrete system is preconditioned in order to enable the use of the principle where “ɛ-uniform stability plus ɛ-uniform consistency implies ɛ-uniform convergence.” Without preconditioning, this principle cannot be applied to convection-diffusion problems because the consistency error is not uniform in ɛ. At the same time, the condition number of the discrete system becomes independent of ɛ due to the same preconditioner; otherwise, the condition number of the discrete system before preconditioning increases when ɛ tends to 0. We obtained such results in an earlier paper, but only for the standard Shishkin mesh. In a nontrivial generalization, we show here that the same proof techniques can be applied to the whole class of Shishkin-type meshes.
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16

Hammachukiattikul, P., E. Sekar, A. Tamilselvan, R. Vadivel, N. Gunasekaran, and Praveen Agarwal. "Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations." Journal of Mathematics 2021 (June 4, 2021): 1–15. http://dx.doi.org/10.1155/2021/6636607.

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In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).
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17

KADALBAJOO, M. K., and ARJUN SINGH YADAW. "PARAMETER-UNIFORM FINITE ELEMENT METHOD FOR TWO-PARAMETER SINGULARLY PERTURBED PARABOLIC REACTION-DIFFUSION PROBLEMS." International Journal of Computational Methods 09, no. 04 (December 2012): 1250047. http://dx.doi.org/10.1142/s0219876212500478.

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In this paper, parameter-uniform numerical methods for a class of singularly perturbed one-dimensional parabolic reaction-diffusion problems with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and finite element method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O(N-2( ln N)2 + Δt). Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.
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18

Vulanović, Relja. "An Almost Sixth-Order Finite-Difference Method for Semilinear Singular Perturbation Problems." Computational Methods in Applied Mathematics 4, no. 3 (2004): 368–83. http://dx.doi.org/10.2478/cmam-2004-0020.

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AbstractThe discretization meshes of the Shishkin type are more suitable for high- order finite-difference schemes than Bakhvalov-type meshes. This point is illustrated by the construction of a hybrid scheme for a class of semilinear singularly perturbed reaction-diffusion problems. A sixth-order five-point equidistant scheme is used at most of the mesh points inside the boundary layers, whereas lower-order three-point schemes are used elsewhere. It is proved under certain conditions that this combined scheme is almost sixth-order accurate and that its error does not increase when the perturbation parameter tends to zero.
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19

Linß, Torsten. "An upwind difference scheme on a novel Shishkin-type mesh for a linear convection–diffusion problem." Journal of Computational and Applied Mathematics 110, no. 1 (October 1999): 93–104. http://dx.doi.org/10.1016/s0377-0427(99)00198-3.

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20

Zahra, W. K., and M. Van Daele. "Discrete Spline Solution of Singularly Perturbed Problem with Two Small Parameters on a Shishkin-Type Mesh." Computational Mathematics and Modeling 29, no. 3 (June 1, 2018): 367–81. http://dx.doi.org/10.1007/s10598-018-9416-3.

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21

Ishwariya, R., J. J. H. Miller, and S. Valarmathi. "A parameter uniform essentially first-order convergent numerical method for a parabolic system of singularly perturbed differential equations of reaction–diffusion type with initial and Robin boundary conditions." International Journal of Biomathematics 12, no. 01 (January 2019): 1950001. http://dx.doi.org/10.1142/s1793524519500013.

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In this paper, a class of linear parabolic systems of singularly perturbed second-order differential equations of reaction–diffusion type with initial and Robin boundary conditions is considered. The components of the solution [Formula: see text] of this system are smooth, whereas the components of [Formula: see text] exhibit parabolic boundary layers. A numerical method composed of a classical finite difference scheme on a piecewise uniform Shishkin mesh is suggested. This method is proved to be first-order convergent in time and essentially first-order convergent in the space variable in the maximum norm uniformly in the perturbation parameters.
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22

Russell, Stephen, and Martin Stynes. "Balanced-norm error estimates for sparse grid finite element methods applied to singularly perturbed reaction–diffusion problems." Journal of Numerical Mathematics 27, no. 1 (March 26, 2019): 37–55. http://dx.doi.org/10.1515/jnma-2017-0079.

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Abstract We consider a singularly perturbed linear reaction–diffusion problem posed on the unit square in two dimensions. Standard finite element analyses use an energy norm, but for problems of this type, this norm is too weak to capture adequately the behaviour of the boundary layers that appear in the solution. To address this deficiency, a stronger so-called ‘balanced’ norm has been considered recently by several researchers. In this paper we shall use two-scale and multiscale sparse grid finite element methods on a Shishkin mesh to solve the reaction–diffusion problem, and prove convergence of their computed solutions in the balanced norm.
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23

Zahra, W. K., and M. Van Daele. "Correction to: Discrete Spline Solution of Singularly Perturbed Problem with Two Small Parameters on a Shishkin-Type Mesh." Computational Mathematics and Modeling 29, no. 4 (October 2018): 475. http://dx.doi.org/10.1007/s10598-018-9427-0.

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24

Kumar, V. "High-Order Compact Finite-Difference Scheme for Singularly-Perturbed Reaction-Diffusion Problems on a New Mesh of Shishkin Type." Journal of Optimization Theory and Applications 143, no. 1 (May 14, 2009): 123–47. http://dx.doi.org/10.1007/s10957-009-9547-y.

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25

Sahu, Subal Ranjan, and Jugal Mohapatra. "Numerical investigation of time delay parabolic differential equation involving two small parameters." Engineering Computations 38, no. 6 (January 20, 2021): 2882–99. http://dx.doi.org/10.1108/ec-07-2020-0369.

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Purpose The purpose of this study is to provide a robust numerical method for a two parameter singularly perturbed delay parabolic initial boundary value problem (IBVP). Design/methodology/approach To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. Here, the authors have used Shishkin type meshes for spatial discretization. Findings It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Originality/value This paper deals with the numerical study of a two parameter singularly perturbed delay parabolic IBVP. To solve the problem, the authors have used a hybrid scheme combining the midpoint scheme, the upwind scheme and the second-order central difference scheme for the spatial derivatives. The backward Euler scheme on a uniform mesh is used to approximate the time derivative. The convergence analysis is carried out. It is observed that the proposed method converges uniformly with almost second-order spatial accuracy with respect to the discrete maximum norm. Numerical experiments illustrate the efficiency of the proposed scheme.
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26

CHRISTY ROJA, J., and A. TAMILSELVAN. "SHOOTING METHOD FOR SINGULARLY PERTURBED FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS OF REACTION-DIFFUSION TYPE." International Journal of Computational Methods 10, no. 06 (May 2, 2013): 1350041. http://dx.doi.org/10.1142/s0219876213500412.

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A class of singularly perturbed boundary value problems (SPBVPs) for fourth-order ordinary differential equations (ODEs) is considered. The SPBVP is reduced into a weakly coupled system of two ODEs subject to suitable initial and boundary conditions. In order to solve them numerically, a method is suggested in which the given interval is divided into two inner regions (boundary layer regions) and one outer region. Two initial-value problems associated with inner regions and one boundary value problem corresponding to the outer region are derived from the given SPBVP. In each of the two inner regions, an initial value problem is solved by using fitted mesh finite difference (FMFD) scheme on Shishkin mesh and the boundary value problem corresponding to the outer region is solved by using classical finite difference (CFD) scheme on Shishkin mesh. A combination of the solution so obtained yields a numerical solution of the boundary value problem on the whole interval. First, in this method, we find the zeroth-order asymptotic expansion approximation of the solution of the weakly coupled system. Error estimates are derived. Examples are presented to illustrate the numerical method. This method is suitable for parallel computing.
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27

Govindarao, Lolugu, and Jugal Mohapatra. "A Second Order Weighted Numerical Scheme on Nonuniform Meshes for Convection Diffusion Parabolic Problems." European Journal of Computational Mechanics, January 14, 2020. http://dx.doi.org/10.13052/ejcm2642-2085.2854.

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In this article, a singularly perturbed parabolic convection-diffusion equation on a rectangular domain is considered. The solution of the problem possesses regular boundary layer which appears in the spatial variable. To discretize the time derivative, we use two type of schemes, first the implicit Euler scheme and second the implicit trapezoidal scheme on a uniform mesh. For approximating the spatial derivatives, we use the monotone hybrid scheme, which is a combination of midpoint upwind scheme and central difference scheme with variable weights on Shishkin-type meshes (standard Shishkin mesh, Bakhvalov-Shishkin mesh and modified Bakhvalov-Shishkin mesh). We prove that both numerical schemes converge uniformly with respect to the perturbation parameter and are of second order accurate. Thomas algorithm is used to solve the tri-diagonal system. Finally, to support the theoretical results, we present a numerical experiment by using the proposed methods.
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28

Chawla, Sheetal, Jagbir Singh, and Urmil. "An Analysis of the Robust Convergent Method for a Singularly Perturbed Linear System of Reaction–Diffusion Type Having Nonsmooth Data." International Journal of Computational Methods, August 25, 2021, 2150056. http://dx.doi.org/10.1142/s0219876221500560.

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In this paper, a coupled system of [Formula: see text] second-order singularly perturbed differential equations of reaction–diffusion type with discontinuous source term subject to Dirichlet boundary conditions is studied, where the diffusive term of each equation is being multiplied by the small perturbation parameters having different magnitudes and coupled through their reactive term. A discontinuity in the source term causes the appearance of interior layers on either side of the point of discontinuity in the continuous solution in addition to the boundary layer at the end points of the domain. Unlike the case of a single equation, the considered system does not obey the maximum principle. To construct a numerical method, a classical finite difference scheme is defined in conjunction with a piecewise-uniform Shishkin mesh and a graded Bakhvalov mesh. Based on Green’s function theory, it has been proved that the proposed numerical scheme leads to an almost second-order parameter-uniform convergence for the Shishkin mesh and second-order parameter-uniform convergence for the Bakhvalov mesh. Numerical experiments are presented to illustrate the theoretical findings.
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29

Yadav, Swati, and Pratima Rai. "A higher order scheme for singularly perturbed delay parabolic turning point problem." Engineering Computations ahead-of-print, ahead-of-print (July 20, 2020). http://dx.doi.org/10.1108/ec-03-2020-0172.

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Purpose The purpose of this study is to construct and analyze a parameter uniform higher-order scheme for singularly perturbed delay parabolic problem (SPDPP) of convection-diffusion type with a multiple interior turning point. Design/methodology/approach The authors construct a higher-order numerical method comprised of a hybrid scheme on a generalized Shishkin mesh in space variable and the implicit Euler method on a uniform mesh in the time variable. The hybrid scheme is a combination of simple upwind scheme and the central difference scheme. Findings The proposed method has a convergence rate of order O(N−2L2+Δt). Further, Richardson extrapolation is used to obtain convergence rate of order two in the time variable. The hybrid scheme accompanied with extrapolation is second-order convergent in time and almost second-order convergent in space up to a logarithmic factor. Originality/value A class of SPDPPs of convection-diffusion type with a multiple interior turning point is studied in this paper. The exact solution of the considered class of problems exhibit two exponential boundary layers. The theoretical results are supported via conducting numerical experiments. The results obtained using the proposed scheme are also compared with the simple upwind scheme.
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