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Srinivasan Natesan - One of the best experts on this subject based on the ideXlab platform.
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optimal error estimate using mesh equidistribution technique for singularly perturbed system of reaction diffusion boundary value problems
Applied Mathematics and Computation, 2014Co-Authors: Srinivasan NatesanAbstract:In this article, we study the problem of determining an appropriate grading of meshes for a system of coupled singularly perturbed reaction-diffusion problems having diffusion parameters with different magnitudes. The central difference scheme is used to discretize the problem on adaptively generated mesh where the mesh equation is derived using an equidistribution principle. An a priori Monitor Function is obtained from the error estimate. A suitable a posteriori analogue of this Monitor Function is also derived for the mesh construction which will lead to an optimal second-order parameter uniform convergence. We present the results of numerical experiments for linear and semilinear reaction-diffusion systems to support the effectiveness of our preferred Monitor Function obtained from theoretical analysis.
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uniformly convergent numerical method for singularly perturbed parabolic initial boundary value problems with equidistributed grids
International Journal of Computer Mathematics, 2014Co-Authors: S Gowrisankar, Srinivasan NatesanAbstract:In this article, we study the numerical solution of singularly perturbed parabolic convection–diffusion problems exhibiting regular boundary layers. To solve these problems, we use the classical upwind finite difference scheme on layer-adapted nonuniform grids. The nonuniform grids are obtained by equidistribution of a positive Monitor Function, which is a linear combination of a constant and the second-order spatial derivative of the singular component of the solution on every temporal level. Truncation error and the stability analysis are obtained. Parameter-uniform error estimates are derived for the numerical solution. To support the theoretical results, numerical experiments are carried out.
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uniformly convergent numerical method for singularly perturbed differential difference equation using grid equidistribution
International Journal for Numerical Methods in Biomedical Engineering, 2010Co-Authors: Jugal Mohapatra, Srinivasan NatesanAbstract:In this paper, a class of singularly perturbed differential-difference equations with small delay and shift terms is considered. A numerical method comprising of upwind finite difference operator on an adaptive grid, which is formed by equidistributing the arc-length Monitor Function, is constructed for approximating the solution. The method is proved to be robust, in the sense that the discrete solution obtained converges in the maximum norm to the exact solution uniformly with respect to the perturbation parameter. Parameter-uniform error bounds for the numerical approximations are established. Numerical examples support the theoretical results. Copyright © 2010 John Wiley & Sons, Ltd.
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uniform convergence analysis of finite difference scheme for singularly perturbed delay differential equation on an adaptively generated grid
Numerical Mathematics-theory Methods and Applications, 2009Co-Authors: Jugal Mohapatra, Srinivasan NatesanAbstract:Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper, we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length Monitor Function. It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter. Numerical experiments illustrate in practice the result of convergence proved theoretically. where 0 0,∀x∈ . Such a problem is sometimes addressed as two-parameter problem. The argument for small delay problems
D M Sloan - One of the best experts on this subject based on the ideXlab platform.
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on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution
Journal of Computational Physics, 2001Co-Authors: G Beckett, J A Mackenzie, Alison Ramage, D M SloanAbstract:Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a Monitor Function and it is one of the moving mesh methods proposed by W. Huang, Y. Ren, and R. D. Russell (1994, SIAM J. Numer. Anal.31 709). We show how the accuracy of the computations is strongly dependent on the choice of Monitor Function, and we present a Monitor Function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.
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numerical solution of a singularly perturbed two point boundary value problem using equidistribution analysis of convergence
Journal of Computational and Applied Mathematics, 2000Co-Authors: D M Sloan, Tao TangAbstract:Abstract Adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near-singular solutions. Adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers (see, for example, Mulholland et al., SIAM J. Sci. Comput. 19(4) (1998) 1261–1289). Much is now being done in developing error analyes for methods that are based on adaptivity. In this paper, we present a rigorous error analysis for the solution of a singularly perturbed two-point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution. The discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a Monitor Function based on arc-length. An error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter, epsilon. The epsilon-uniform convergence is confirmed by numerical computations.
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analysis of difference approximations to a singularly perturbed two point boundary value problem on an adaptively generated grid
Journal of Computational and Applied Mathematics, 1999Co-Authors: Y Qui, D M SloanAbstract:Over the last few years there has been a significant growth in the use of adaptive grid methods for the numerical solution of differential equations with steep solutions. Little has been done, however, on the error analysis of adaptive methods. In this paper, we present an analysis for an upwind finite difference solution of a singular perturbation problem on a grid that is generated adaptively by equidistributing a Monitor Function based on the exact solution. It is shown that the discrete solutions converge uniformly with respect to the perturbation parameter, epsilon. This epsilon-uniform convergence is illustrated by numerical computations.
G Beckett - One of the best experts on this subject based on the ideXlab platform.
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uniformly convergent high order finite element solutions of a singularly perturbed reaction diffusion equation using mesh equidistribution
Applied Numerical Mathematics, 2001Co-Authors: G Beckett, J A MackenzieAbstract:We study the numerical approximation of a singularly perturbed reaction-diffusion equation using a pth order Galerkin finite element method on a non-uniform grid. The grid is constructed by equidistributing a strictly positive Monitor Function which is a linear combination of a constant floor and a power of the second derivative of a representation of the boundary layers-obtained using a suitable decomposition of the analytical solution. By the appropriate selection of the Monitor Function parameters we prove that the numerical solution is insensitive to the size of the singular perturbation parameter and achieves the optimal rate of convergence with respect to the mesh density.
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on a uniformly accurate finite difference approximation of a singulary peturbed reaction diffusion problem using grid equidistribution
Journal of Computational and Applied Mathematics, 2001Co-Authors: G Beckett, J A MackenzieAbstract:We examine the convergence properties of a finite difference approximation of a singularly perturbed reaction-diffusion boundary value problem using a nonuniform grid. The grid is based on the equidistribution of a positive Monitor Function that is a linear combination of a constant floor and a power of the second derivative of the solution. Analysis shows how the Monitor Function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter. The use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids. Numerical results are given that confirm the uniform convergence rates.
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on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution
Journal of Computational Physics, 2001Co-Authors: G Beckett, J A Mackenzie, Alison Ramage, D M SloanAbstract:Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a Monitor Function and it is one of the moving mesh methods proposed by W. Huang, Y. Ren, and R. D. Russell (1994, SIAM J. Numer. Anal.31 709). We show how the accuracy of the computations is strongly dependent on the choice of Monitor Function, and we present a Monitor Function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.
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convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem
Applied Numerical Mathematics, 2000Co-Authors: G Beckett, J A MackenzieAbstract:Abstract We derive e -uniform error estimates for two first-order upwind discretizations of a model inhomogeneous, second-order, singularly perturbed boundary value problem on a non-uniform grid. Here, e is the small parameter multiplying the highest derivative term. The grid is suggested by the equidistribution of a positive Monitor Function which is a linear combination of a constant floor and a power of the second derivative of the solution. Our analysis shows how the floor should be chosen to ensure e -uniform convergence and indicates the convergence behaviour for such grids. Numerical results are presented which confirm the e -uniform convergence rates.
J A Mackenzie - One of the best experts on this subject based on the ideXlab platform.
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uniformly convergent high order finite element solutions of a singularly perturbed reaction diffusion equation using mesh equidistribution
Applied Numerical Mathematics, 2001Co-Authors: G Beckett, J A MackenzieAbstract:We study the numerical approximation of a singularly perturbed reaction-diffusion equation using a pth order Galerkin finite element method on a non-uniform grid. The grid is constructed by equidistributing a strictly positive Monitor Function which is a linear combination of a constant floor and a power of the second derivative of a representation of the boundary layers-obtained using a suitable decomposition of the analytical solution. By the appropriate selection of the Monitor Function parameters we prove that the numerical solution is insensitive to the size of the singular perturbation parameter and achieves the optimal rate of convergence with respect to the mesh density.
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on a uniformly accurate finite difference approximation of a singulary peturbed reaction diffusion problem using grid equidistribution
Journal of Computational and Applied Mathematics, 2001Co-Authors: G Beckett, J A MackenzieAbstract:We examine the convergence properties of a finite difference approximation of a singularly perturbed reaction-diffusion boundary value problem using a nonuniform grid. The grid is based on the equidistribution of a positive Monitor Function that is a linear combination of a constant floor and a power of the second derivative of the solution. Analysis shows how the Monitor Function can be chosen to ensure that the accuracy of the numerical approximation is insensitive to the size of the singular perturbation parameter. The use of equidistribution principles appears in many practical grid adaption schemes and our analysis provides insight into the convergence behaviour on such grids. Numerical results are given that confirm the uniform convergence rates.
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on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution
Journal of Computational Physics, 2001Co-Authors: G Beckett, J A Mackenzie, Alison Ramage, D M SloanAbstract:Numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one-dimensional partial differential equations (PDEs). The particular method considered here is an adaptive finite difference method based on the equidistribution of a Monitor Function and it is one of the moving mesh methods proposed by W. Huang, Y. Ren, and R. D. Russell (1994, SIAM J. Numer. Anal.31 709). We show how the accuracy of the computations is strongly dependent on the choice of Monitor Function, and we present a Monitor Function that yields an optimal rate of convergence. Motivated by efficiency considerations for problems in two or more space dimensions, we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical PDE. The accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time-step control mechanism is integrated into the scheme.
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convergence analysis of finite difference approximations on equidistributed grids to a singularly perturbed boundary value problem
Applied Numerical Mathematics, 2000Co-Authors: G Beckett, J A MackenzieAbstract:Abstract We derive e -uniform error estimates for two first-order upwind discretizations of a model inhomogeneous, second-order, singularly perturbed boundary value problem on a non-uniform grid. Here, e is the small parameter multiplying the highest derivative term. The grid is suggested by the equidistribution of a positive Monitor Function which is a linear combination of a constant floor and a power of the second derivative of the solution. Our analysis shows how the floor should be chosen to ensure e -uniform convergence and indicates the convergence behaviour for such grids. Numerical results are presented which confirm the e -uniform convergence rates.
Natalia Kopteva - One of the best experts on this subject based on the ideXlab platform.
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a robust grid equidistribution method for a one dimensional singularly perturbed semilinear reaction diffusion problem
Ima Journal of Numerical Analysis, 2011Co-Authors: Naresh M Chadha, Natalia KoptevaAbstract:The numerical solution of a singularly perturbed semilinear reaction-diffusion two-point boundary-value problem is addressed. The method considered is adaptive movement of a fixed number (N + 1) of mesh points by equidistribution of a Monitor Function that uses discrete second-order derivatives. We extend the analysis by Kopteva & Stynes (2001, SIAM J. Numer. Anal., 39, 1446-1467) to a new equation and a more intricate Monitor Function. It is proved that there exists a solution to the fully discrete equidistribution problem, i.e. a mesh exists that equidistributes the discrete Monitor Function computed from the discrete solution on this mesh. Furthermore, in the case when the boundary-value problem is linear, it is shown that after O(|1ne|/ In N) iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves second-order accuracy in the maximum norm, uniformly in the diffusion coefficient e 2 . Numerical experiments are presented that support our theoretical results.
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maximum norm a posteriori error estimates for a 1d singularly perturbed semilinear reaction diffusion problem
Ima Journal of Numerical Analysis, 2006Co-Authors: Natalia KoptevaAbstract:A singularly perturbed semilinear two-point boundary-value problem is discretized on arbitrary non-uniform meshes. We present second-order maximum norm a posteriori error estimates that hold true uniformly in the small parameter. Their application to Monitor-Function equidistribution and a posteriori mesh refinement are discussed. Numerical results are presented that support our theoretical estimates.