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Kailash C. Patidar - One of the best experts on this subject based on the ideXlab platform.
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layer resolving Fitted Mesh method for parabolic convection diffusion problems with a variable diffusion
Journal of Applied Mathematics and Computing, 2021Co-Authors: Charles K Mbayi, Justin B Munyakazi, Kailash C. PatidarAbstract:In this paper, we constructed a Fitted Mesh finite difference method for solving a class of time-dependent singularly perturbed turning point convection-diffusion problems whose solution exhibits an interior layer. The diffusion coefficient in the underlying PDE is a quadratic function of the space variable and contains a perturbation parameter. While such problems have been studied in the case of boundary layers, little has been achieved for interior layer problems where the coefficient functions are considered to be dependent on the space variable alone. In this work, we focus our attention to such problems where the coefficient functions are dependent of both the space and time variables. Following the work of Liseikin (USSR Computational Mathematics and Mathematical Physics 26(6), 133–139, 1986), we establish bounds on the solution and its derivatives. Then we discretize the time derivative using an implicit Euler method. This discretization results in a set of two-point boundary value problems (TPBVPs). We then construct a Fitted Mesh finite difference method to solve these TPBVPs. This method is analyzed for stability and convergence. We proved that it satisfies a minimum principle and is uniformly convergent with respect to the perturbation parameter. In order to improve the accuracy of the proposed method, we use the Richardson extrapolation. Finally, we present some numerical experiments to validate our theoretical findings.
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a novel Fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation
Applied Mathematics and Computation, 2011Co-Authors: Eihab B M Bashier, Kailash C. PatidarAbstract:Abstract We design a robust Fitted operator finite difference method for the numerical solution of a singularly perturbed delay parabolic partial differential equation. This method is unconditionally stable and is convergent with order O ( k + h 2 ) , where k and h are respectively the time and space step-sizes, which is better than the one obtained by Ansari et al. [A.R. Ansari, S.A. Bakr, G.I. Shishkin, A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations, J. Comput. Appl. Math. 205 (2007) 552–566] where they have used a Fitted Mesh finite difference method. Their method was of the order O N t - 1 + N x - 2 ln 2 N x , where Nt and Nx denote the total number of sub-intervals in the time and space directions. The performance of our method is illustrated through some numerical experiments. We also compare our results with those obtained by a standard finite difference method as well as other works seen in the literature. In addition, we provide a novel proof for the bounds on partial derivatives of the solution of the continuous problem.
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an almost second order Fitted Mesh numerical method for a singularly perturbed delay parabolic partial differential equation
Neural Parallel & Scientific Computations archive, 2010Co-Authors: Eihab B M Bashier, Kailash C. PatidarAbstract:In this paper we develop a numerical method for solving a singularly perturbed delay parabolic partial differential equation. The proposed method consists of Crank-Nicolson finite difference method constructed on a Mesh of Shishkin type and hence referred to as a Fitted Mesh finite difference method. We analyzed the method for stability and convergence and found that it is unconditionally stable and converges with order O (Nt-2 + Nx-2ln2Nx) where Nt and Nx are the numbers of subintervals in the t and x directions, respectively. The performance of the method is illustrated through numerical experiments.
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a new class of Fitted Mesh finite element method for convection diffusion reaction problems
Neural Parallel & Scientific Computations archive, 2010Co-Authors: Kailash C. PatidarAbstract:We design a novel finite element method for a class of singularly perturbed two-point boundary value problems. Using the intrinsic singular feature of the solution and some a priori error estimates, we design the method, based upon a finite element discretization on a suitably refined Mesh. This new method is referred as Mesh Refinement Finite Element Method and is proved to be e-uniformly convergent in appropriate norms. Optimal values of the Mesh generating parameter (referred to as v) is obtained via a priori error analysis. These optimal values are then considered as main indicators in identifying a fixed value of v that can provide reliable Mesh on which the finite element method provides parameter robust numerical results.
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limitations of richardson s extrapolation for a high order Fitted Mesh method for self adjoint singularly perturbed problems
Journal of Applied Mathematics and Computing, 2010Co-Authors: Justin B Munyakazi, Kailash C. PatidarAbstract:The aim of this paper is to investigate whether we can accelerate the order of convergence of existing high order methods to solve some singularly perturbed two-point BVPs. To this end, we consider a Fitted Mesh finite difference method of Patidar (Appl. Math. Comput., 188:720–733, 2007) applied on a Mesh of Shishkin type for the solution of self-adjoint problem which is e-uniformly convergent of order four. We attempted to increase the order of convergence by Richardson’s extrapolation and discovered that this well-known convergence acceleration technique has some limitations. We observe that even though this extrapolation technique improves the accuracy slightly, it does not increase the rate of convergence which is originally four for the underlying method for the problem above. Theoretical investigations are demonstrated by some numerical experiments.
V Shanthi - One of the best experts on this subject based on the ideXlab platform.
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Fitted Mesh method for a weakly coupled system of singularly perturbed reaction convection diffusion problems with discontinuous source term
Ain Shams Engineering Journal, 2016Co-Authors: Pathan Mahabub Basha, V ShanthiAbstract:Abstract In this article, a parameter-uniform numerical method for a weakly coupled system of singularly perturbed reaction–convection–diffusion problems with discontinuous source term containing two small parameters multiplied to the highest and second highest derivative is presented. A Fitted Mesh method using an upwind finite difference scheme on piecewise-uniform Shishkin Mesh is constructed. Error analysis is undertaken and numerical results are provided to support the theoretical error bounds.
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Fitted Mesh method for singularly perturbed robin type boundary value problem with discontinuous source term
International Journal of Applied and Computational Mathematics, 2015Co-Authors: M Chandru, V ShanthiAbstract:In this paper, second order singularly perturbed convection-diffusion Robin type problem with a discontinuous source term is considered. Due to the discontinuity interior layers appears in the solution. A numerical method is constructed for this problem which involves an appropriate piecewise—uniform Mesh for the boundary and interior layers. The method is shown to be parameter uniformly convergent with respect to the singular perturbation parameter. Numerical examples are presented to illustrate the theoretical results.
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Fitted Mesh method for singularly perturbed reaction-convection-diffusion problems with boundary and interior layers
Journal of Applied Mathematics and Computing, 2006Co-Authors: V Shanthi, N Ramanujam, S. NatesanAbstract:A robust numerical method for a singularly perturbed secondorder 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.
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computational methods for reaction diffusion problems for fourth order ordinary differential equations with a small parameter at the highest derivative
Applied Mathematics and Computation, 2004Co-Authors: V Shanthi, N RamanujamAbstract:In this paper basically-asymptotic numerical methods for solving singularly perturbed two-point boundary value problems for fourth order ordinary differential equations of the form-@ey^i^v(x)+b(x)y^'^'(x)+c(x)y(x)=f(x),x@?D:=(0,1),y(0)=p,y^'(0)=q,y^'^'(0)=r,y^'^'(1)=s,is considered. Here a prime ''''' denotes a differentiation with respect to x, b(x), c(x) and f(x) are smooth functions, b(x)>=@b>0, 0>=c(x)>=-@c, @c>0 and 0<@e@?1. The above boundary value problem is transformed into an equivalent weakly coupled system of two first order ordinary differential equations subject to suitable initial conditions and one second order singularly perturbed ordinary differential equations subject to suitable boundary conditions. In order to solve this system three computational methods are suggested in this paper. In these methods, first we find a zero order asymptotic approximation of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero order asymptotic approximation of the solution in the second order equation. Then the second order equation is solved separately by three methods namely Fitted operator method, Fitted Mesh method and boundary value technique. Error estimates are derived and examples are provided to illustrate the methods.
John J H Miller - One of the best experts on this subject based on the ideXlab platform.
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parameter uniform essentially first order convergence of a Fitted Mesh method for a class of parabolic singularly perturbed robin problem for a system of reaction diffusion equations
arXiv: Numerical Analysis, 2019Co-Authors: R Ishwariya, John J H Miller, S ValarmathiAbstract: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 $\vec u$ of this system exhibit parabolic boundary layers with sublayers. 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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an experimental technique for computing parameter uniform error estimates for numerical solutions of singular perturbation problems with an application to prandtl s problem at high reynolds number
Applied Numerical Mathematics, 2002Co-Authors: Paul A Farrell, Alan F Hegarty, John J H Miller, Eugene Oriordan, G I ShishkinAbstract:In this paper we describe an experimental technique for computing realistic values of the parameter-uniform order of convergence and error constant in the maximum norm associated with a parameter-uniform numerical method for solving singularly perturbed problems. We employ the technique to compute Reynolds-uniform error bounds in the maximum norm for the numerical solutions generated by a Fitted-Mesh upwind finite difference method applied to Prandtl's problem arising from laminar flow past a thin flat plate. Thus we illustrate the efficiency of the technique for finding realistic parameter-uniform error bounds in the maximum norm for the approximate solutions generated by numerical methods for which no theoretical error analysis is available.
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parameter uniform Fitted Mesh method for quasilinear differential equations with boundary layers 1
Computational Methods in Applied Mathematics Comput, 2001Co-Authors: Paul A Farrell, John J H MillerAbstract:Singularly perturbed quasilinear boundary value problems exhibiting boundary layers are considered. Special piecewise-uniform Meshes are constructed which are Fitted to these boundary layers. Numerical methods composed of upwind dierence operators and these Fitted Meshes are shown to be parameter robust, in the sense that the solutions satisfy an error estimate in the maximum norm which is independent of the value of the singular perturbation parameter. Numerical results supporting the theory are presented. 2000 Mathematics Subject Classification: 34E10; 65L10. Keywords: quasilinear boundary value problem, singular perturbation, finite- dierence method, piecewise uniform Mesh, "-uniform error estimate.
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Fitted Mesh methods for problems with parabolic boundary layers
1998Co-Authors: John J H Miller, Gregori Shishkin, L P Shishkina, P M QuinlanAbstract:A Dirichlet boundary value problem for a linear parabolic dierential equation is studied on a rectangular domain in the x t plane. The coecient of the second order space derivative is a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. It is proved that a numerical method, comprising a standard nite dierence operator (centred in space, implicit in time) on a tted piecewise uniform Mesh of NxNt elements condensing in the boundary layers, is uniform with respect to the small parameter, in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the semi-open interval (0,1]. More specically, it is shown that the errors are bounded in the maximum norm by C((N 1 x lnNx) 2 +N 1 t ), where C is a constant independent not only of Nx and Nt but also of the small parameter. Numerical results are presented, which validate numerically this theoretical result and show that a numerical method consisting of the same nite dierence operator on a uniform Mesh of NxNt elements is not uniform with respect to the small parameter.
S. Natesan - One of the best experts on this subject based on the ideXlab platform.
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Fitted Mesh method for singularly perturbed reaction-convection-diffusion problems with boundary and interior layers
Journal of Applied Mathematics and Computing, 2006Co-Authors: V Shanthi, N Ramanujam, S. NatesanAbstract:A robust numerical method for a singularly perturbed secondorder 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.
Mohan K. Kadalbajoo - One of the best experts on this subject based on the ideXlab platform.
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comparative study of singularly perturbed two point bvps via Fitted Mesh finite difference method b spline collocation method and finite element method
Applied Mathematics and Computation, 2008Co-Authors: Mohan K. Kadalbajoo, Arjun Singh Yadaw, Devendra KumarAbstract:Abstract The objective of this paper is to present a comparative study of Fitted-Mesh finite difference method, B-spline collocation method and finite element method for general singularly perturbed two-point boundary value problems. Due to the small parameter ϵ , the boundary layer arises. We have taken a piecewise-uniform Fitted-Mesh to resolve the boundary layer and we have shown that Fitted-Mesh finite difference method has ϵ -uniform first order convergence, B-spline collocation method has almost second order ϵ -uniform convergence and Ritz–Galerkin method also has almost second order ϵ -uniform convergence. Two test examples have been solved to compare the maximum absolute error and rate of convergence of the methods.
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Fitted Mesh b spline collocation method for singularly perturbed differential difference equations with small delay
Applied Mathematics and Computation, 2008Co-Authors: Mohan K. Kadalbajoo, Devendra KumarAbstract:Abstract This paper deals with the singularly perturbed boundary value problem for a linear second order differential–difference equation of the convection–diffusion type with small delay parameter δ of o ( e ) whose solution has a boundary layer. The Fitted Mesh technique is employed to generate a piecewise-uniform Mesh, condensed in the neighborhood of the boundary layers. B-spline collocation method is used with Fitted Mesh. Parameter-uniform convergence analysis of the method is discussed. The method is shown to have almost second order parameter-uniform convergence. The effect of small delay δ on boundary layer has also been discussed. Several examples are considered to demonstrate the performance of the proposed scheme and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution.
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e uniformly convergent Fitted methods for the numerical solution of the problems arising from singularly perturbed general ddes
Applied Mathematics and Computation, 2006Co-Authors: Mohan K. Kadalbajoo, Kailash C. PatidarAbstract: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.
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e uniformly convergent Fitted Mesh finite difference methods for general singular perturbation problems
Applied Mathematics and Computation, 2006Co-Authors: Mohan K. Kadalbajoo, Kailash C. PatidarAbstract: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.
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b spline method for solving general singularly perturbed boundary value problems using Fitted Mesh
Computing Letters, 2006Co-Authors: Mohan K. Kadalbajoo, Vivek K AggarwalAbstract:In this paper we develop B-spline method for solving a class of Singularly Perturbed two point boundary value problems given asLy = ey″ = F(x,y,y′), x ∈ (0,1) (1)y(0) = ν0 y(1) = ν1, ν0,ν1 ∈ R (2)We use the Fitted Mesh technique to generate piecewise uniform Mesh, and use B-spline method which leads to a tridiagonal linear system. In case of non-linear problems we first linearize the equation using Quasilinearization technique and the resulting problem is solved by B-spline. The convergence analysis is given and the method is shown to have uniform convergence. Numerical illustrations are given in the end to demonstrate the efficiency of our method.