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C. Clavero - One of the best experts on this subject based on the ideXlab platform.

  • An efficient and Uniformly Convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type
    Numerical Algorithms, 2019
    Co-Authors: C. Clavero, J. C. Jorge
    Abstract:

    In this work we are interested in the numerical approximation of the solutions to 1D semilinear parabolic singularly perturbed systems of reaction-diffusion type, in the general case where the diffusion parameters for each equation can have different orders of magnitude. The numerical method combines the classical central finite differences scheme to discretize in space and a linearized fractional implicit Euler method together with a splitting by components technique to integrate in time. In this way, only tridiagonal linear systems must be solved to compute the numerical solution; consequently, the computational cost of the algorithm is considerably less than that of classical schemes. If the spatial discretization is defined on appropriate nonuniform meshes, the method is Uniformly Convergent of first order in time and almost second order in space. Numerical results for some test problems are presented which corroborate in practice the uniform convergence and the efficiency of the algorithm.

  • Uniformly Convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type
    Numerical Algorithms, 2019
    Co-Authors: C. Clavero, José Luis Gracia
    Abstract:

    In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space Uniformly Convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques.

  • An improved Uniformly Convergent scheme in space for 1D parabolic reaction-diffusion systems
    Applied Mathematics and Computation, 2014
    Co-Authors: C. Clavero, José Luis Gracia
    Abstract:

    In this paper the numerical approximation of 1D parabolic singularly perturbed systems with two equations of reaction-diffusion type is considered. These problems typically exhibit two overlapping boundary layers at both end points of the spatial domain. A decomposition of the exact solution into its regular and singular part is established, given appropriate bounds for the partial derivatives of the exact solution up to sixth order. These bounds are crucial to prove the uniform convergence of a numerical method that combines the classical backward Euler method and a hybrid finite difference scheme defined on a special nonuniform mesh condensing in the layer regions. The numerical method is Uniformly Convergent in the discrete maximum norm, and it has first and third order of convergence in time and space, respectively. Numerical results for some test problems are showed, illustrating in practice the order of convergence theoretically proved.

  • A higher order Uniformly Convergent method with Richardson extrapolation in time for singularly perturbed reaction-diffusion parabolic problems
    Journal of Computational and Applied Mathematics, 2013
    Co-Authors: C. Clavero, José Luis Gracia
    Abstract:

    a b s t r a c t In this paper, we are interested in solving efficiently an initial-boundary value singularly perturbed time-dependent problem of reaction–diffusion type. On a priori special mesh we construct a high order Uniformly Convergent finite difference scheme which combines the implicit Euler method to discretize in time, together with the Richardson extrapolation technique, and a HODIE scheme to discretize in space. The analysis of the uniform convergence splits completely the contribution to the global error of both the time and the space discretizations. We show numerical results for different test problems confirming in practice the order of uniform convergence proved.

  • A robust second-order numerical method for global solution and global normalized flux of singularly perturbed self-adjoint boundary-value problems
    International Journal of Computer Mathematics, 2009
    Co-Authors: C. Clavero, Rajesh K. Bawa, Srinivasan Natesan
    Abstract:

    In this paper, we consider the finite difference hybrid scheme constructed by Natesan et al. for obtaining Uniformly Convergent global solution and Uniformly Convergent normalized flux for self-adjoint singularly perturbed boundary value problems. The global solution is obtained from the numerical solution at the mesh points of this scheme, having almost second-order uniform convergence at the nodal points when it is constructed on a piecewise uniform Shishkin mesh. Using a classical cubic spline, we define the solution and the normalized flux on the entire domain. We prove that the uniform order of convergence of the global solution is the same as that of the hybrid scheme at the mesh points. In addition, the global normalized flux is also almost second-order Uniformly Convergent in the whole domain. We provide theoretical error bounds and some numerical examples showing the efficiency of the proposed technique for obtaining the global solution and the normalized flux.

Kapil K. Sharma - One of the best experts on this subject based on the ideXlab platform.

  • e Uniformly Convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general ddes
    Applied Mathematics and Computation, 2006
    Co-Authors: Mohan K. Kadalbajoo, Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    We consider some problems arising from singularly perturbed general differential difference equations. First we construct (in a new way) and analyze a ''fitted operator finite difference method (FOFDM)'' which is first order @e-Uniformly Convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the ''fitted mesh finite difference method (FMFDM)''. This FMFDM is a direct method and @e-Uniformly Convergent with the nodal error as O(n^-^2ln^2n) which is an improvement over the existing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson's extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as O(n^-^1lnn) where n denotes the total number of sub-intervals of [0,1]. Comparative numerical results are presented in support of the theory.

  • Uniformly Convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance
    International Journal for Numerical Methods in Engineering, 2006
    Co-Authors: Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    A new class of fitted operator finite difference methods are constructed via non-standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second-order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be e-Uniformly Convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.

  • e Uniformly Convergent non standard finite difference methods for singularly perturbed differential difference equations with small delay
    Applied Mathematics and Computation, 2006
    Co-Authors: Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-Uniformly Convergent.

  • ε-Uniformly Convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs
    Applied Mathematics and Computation, 2006
    Co-Authors: Mohan K. Kadalbajoo, Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    We consider some problems arising from singularly perturbed general differential difference equations. First we construct (in a new way) and analyze a ''fitted operator finite difference method (FOFDM)'' which is first order @e-Uniformly Convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the ''fitted mesh finite difference method (FMFDM)''. This FMFDM is a direct method and @e-Uniformly Convergent with the nodal error as O(n^-^2ln^2n) which is an improvement over the existing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson's extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as O(n^-^1lnn) where n denotes the total number of sub-intervals of [0,1]. Comparative numerical results are presented in support of the theory.

  • ε-Uniformly Convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay
    Applied Mathematics and Computation, 2006
    Co-Authors: Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-Uniformly Convergent.

Yunhui Yin - One of the best experts on this subject based on the ideXlab platform.

Kailash C. Patidar - One of the best experts on this subject based on the ideXlab platform.

  • e Uniformly Convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general ddes
    Applied Mathematics and Computation, 2006
    Co-Authors: Mohan K. Kadalbajoo, Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    We consider some problems arising from singularly perturbed general differential difference equations. First we construct (in a new way) and analyze a ''fitted operator finite difference method (FOFDM)'' which is first order @e-Uniformly Convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the ''fitted mesh finite difference method (FMFDM)''. This FMFDM is a direct method and @e-Uniformly Convergent with the nodal error as O(n^-^2ln^2n) which is an improvement over the existing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson's extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as O(n^-^1lnn) where n denotes the total number of sub-intervals of [0,1]. Comparative numerical results are presented in support of the theory.

  • e Uniformly Convergent fitted mesh finite difference methods for general singular perturbation problems
    Applied Mathematics and Computation, 2006
    Co-Authors: Mohan K. Kadalbajoo, Kailash C. Patidar
    Abstract:

    Abstract We consider general singular perturbation problems of the form cey″(x) + a(x)y′(x) + b(x)y(x) = f(x), x ∈ [0, 1]; y(0) = η0, y(1) = η1 with ce equals to both +e and −e, a(x), b(x), f(x) are positive throughout the interval and η 0 , η 1 ∈ R . The indirect methods (those which do not use any acceleration of convergence techniques, e.g., Richardson’s extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type lead the error as O ( n - 1 ln n ) where n denotes the total number of sub-intervals of [0, 1]. In this paper, we systematically describe, a very simple and direct method which reduces the error to O ( n - 2 ln 2 n ) . This method is proved to be e-Uniformly Convergent with the above error bounds, on a piecewise uniform mesh of Shishkin type. The motivation for using this Shishkin mesh is inspired by the quotation of Stynes [M. Stynes, A jejune heuristic mesh theorem, Comput. Methods Appl. Math. 3 (2003) 488–492]: Miller has moved from [J.J.H. Miller, Construction of a FEM for a singularly perturbed problem in 2 dimensions, in: Numerische Behandlung von Differentialgleichungen, Band 2 (Tagung, Math. Forschungsinst., Oberwolfach, 1975), Internat. Ser. Numer. Math., Birkhauser, Basel, vol. 31, 1976, pp. 165–169] the question “what scheme should one use on a given mesh?” to [P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust Computational Techniques for Boundary Layers, Chapman & Hall/CRC, New York, 2000] “what mesh should one use with a given scheme?” The theoretical estimates have been justified by several numerical examples.

  • Uniformly Convergent non-standard finite difference methods for singularly perturbed differential-difference equations with delay and advance
    International Journal for Numerical Methods in Engineering, 2006
    Co-Authors: Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    A new class of fitted operator finite difference methods are constructed via non-standard finite difference methods ((NSFDM)s) for the numerical solution of singularly perturbed differential difference equations having both delay and advance arguments. The main idea behind the construction of our method(s) is to replace the denominator function of the classical second-order derivative with a positive function derived systematically in such a way that it captures significant properties of the governing differential equation and thus provides the reliable numerical results. Unlike other FOFDMs constructed in standard ways, the methods that we present in this paper are fairly simple to construct (and thus enrich the class of fitted operator methods by adding these new methods). These methods are shown to be e-Uniformly Convergent with order two which is the highest possible order of convergence obtained via any fitted operator method for the problems under consideration. This paper further clarifies several doubts, e.g. why a particular scheme is not suitable for the whole range of values of the associated parameters and what could be the possible remedies. Finally, we provide some numerical examples which illustrate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.

  • e Uniformly Convergent non standard finite difference methods for singularly perturbed differential difference equations with small delay
    Applied Mathematics and Computation, 2006
    Co-Authors: Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    Abstract Non-standard finite difference methods (NSFDMs), now-a-days, are playing an important role in solving the real life problems governed by ODEs and/or by PDEs. Many differential models of sciences and engineerings for which the existing methodologies do not give reliable results, these NSFDMs are solving them competitively. To this end, in this paper we consider, second order, linear, singularly perturbed differential difference equations. Using the second of the five non-standard modeling rules of Mickens [R.E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994], the new finite difference methods are obtained for the particular cases of these problems. This rule suggests us to replace the denominator function of the classical second order derivative with a positive function derived systematically in such a way that it captures most of the significant properties of the governing differential equation(s). Both theoretically and numerically, we show that these NSFDMs are e-Uniformly Convergent.

  • ε-Uniformly Convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs
    Applied Mathematics and Computation, 2006
    Co-Authors: Mohan K. Kadalbajoo, Kailash C. Patidar, Kapil K. Sharma
    Abstract:

    We consider some problems arising from singularly perturbed general differential difference equations. First we construct (in a new way) and analyze a ''fitted operator finite difference method (FOFDM)'' which is first order @e-Uniformly Convergent. With the aim of having just one function evaluation at each step, attempts have been made to derive a higher order method via Shishkin mesh to which we refer as the ''fitted mesh finite difference method (FMFDM)''. This FMFDM is a direct method and @e-Uniformly Convergent with the nodal error as O(n^-^2ln^2n) which is an improvement over the existing direct methods (i.e., those which do not use any acceleration of convergence techniques, e.g., Richardson's extrapolation or defect correction, etc.) for such problems on a mesh of Shishkin type that lead the error as O(n^-^1lnn) where n denotes the total number of sub-intervals of [0,1]. Comparative numerical results are presented in support of the theory.

J. C. Jorge - One of the best experts on this subject based on the ideXlab platform.

  • An efficient and Uniformly Convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type
    Numerical Algorithms, 2019
    Co-Authors: C. Clavero, J. C. Jorge
    Abstract:

    In this work we are interested in the numerical approximation of the solutions to 1D semilinear parabolic singularly perturbed systems of reaction-diffusion type, in the general case where the diffusion parameters for each equation can have different orders of magnitude. The numerical method combines the classical central finite differences scheme to discretize in space and a linearized fractional implicit Euler method together with a splitting by components technique to integrate in time. In this way, only tridiagonal linear systems must be solved to compute the numerical solution; consequently, the computational cost of the algorithm is considerably less than that of classical schemes. If the spatial discretization is defined on appropriate nonuniform meshes, the method is Uniformly Convergent of first order in time and almost second order in space. Numerical results for some test problems are presented which corroborate in practice the uniform convergence and the efficiency of the algorithm.

  • A high order Uniformly Convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems
    Numerische Mathematik, 2007
    Co-Authors: B. Bujanda, C. Clavero, J L Gracia, J. C. Jorge
    Abstract:

    In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference scheme constructed on a tailored mesh. We prove that the resulting scheme is ε-Uniformly Convergent of second order in time and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon, which is usually produced in the time integration process when the boundary conditions are time dependent.

  • a high order Uniformly Convergent alternating direction scheme for time dependent reaction diffusion singularly perturbed problems
    Numerische Mathematik, 2007
    Co-Authors: B. Bujanda, C. Clavero, J L Gracia, J. C. Jorge
    Abstract:

    In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference scheme constructed on a tailored mesh. We prove that the resulting scheme is e-Uniformly Convergent of second order in time and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon, which is usually produced in the time integration process when the boundary conditions are time dependent.

  • A high order Uniformly Convergent alternating direction scheme for time dependent reaction–diffusion singularly perturbed problems
    Numerische Mathematik, 2007
    Co-Authors: B. Bujanda, José Luis Gracia, C. Clavero, J. C. Jorge
    Abstract:

    In this work we design and analyze an efficient numerical method to solve two dimensional initial-boundary value reaction–diffusion problems, for which the diffusion parameter can be very small with respect to the reaction term. The method is defined by combining the Peaceman and Rachford alternating direction method to discretize in time, together with a HODIE finite difference scheme constructed on a tailored mesh. We prove that the resulting scheme is e-Uniformly Convergent of second order in time and of third order in spatial variables. Some numerical examples illustrate the efficiency of the method and the orders of uniform convergence proved theoretically. We also show that it is easy to avoid the well-known order reduction phenomenon, which is usually produced in the time integration process when the boundary conditions are time dependent.

  • A Uniformly Convergent alternating direction HODIE finite difference scheme for 2D time-dependent convection–diffusion problems
    IMA Journal of Numerical Analysis, 2005
    Co-Authors: C. Clavero, José Luis Gracia, J. C. Jorge
    Abstract:

    In this work we design and analyze a finite difference scheme used to solve some 2D time dependent convection-diffusion problems, for which we suppose that the convection term is positive in both spatial directions. We use the Peaceman & Rachford method to discretize in time such problems and a HODIE finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize in space. We prove that the method is Uniformly Convergent with respect to the diffusion parameter, reaching almost order two in space. We present some numerical examples which illustrate such behaviour; they show that the numerical method is also suitable in a wider set of singularly perturbed problems that the ones defined by the theoretical restrictions. 2000 Mathematics Subject Classification: 65N12; 65N30; 65N06.