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Srinivasan Natesan - One of the best experts on this subject based on the ideXlab platform.
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parameter uniform numerical method for singularly perturbed 2d delay parabolic convection diffusion problems on shishkin mesh
Journal of Applied Mathematics and Computing, 2019Co-Authors: Abhishek Das, Srinivasan NatesanAbstract:In this article, we study the numerical solution of a singularly perturbed 2D delay parabolic convection–diffusion problem. First, we discretize the domain with a uniform mesh in the Temporal Direction and a special mesh in the spatial Directions. The numerical scheme used to discretize the continuous problem, consists of the implicit-Euler scheme for the time derivative and the classical upwind scheme for the spatial derivatives. Stability analysis is carried out, and parameter-uniform error estimates are derived. The proposed scheme is of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are carried out to verify the theoretical results.
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uniformly convergent numerical method for singularly perturbed 2d delay parabolic convection diffusion problems on bakhvalov shishkin mesh
International Journal of Mathematical Modelling and Numerical Optimisation, 2018Co-Authors: Abhishek Das, Srinivasan NatesanAbstract:In this paper, we consider a class of singularly perturbed 2D delay parabolic convection-diffusion initial-boundary-value problems. To solve this problem numerically, we use a modified Shishkin mesh (Bakhvalov-Shishkin mesh) for the discretisation of the domain in the spatial Directions and uniform mesh in the Temporal Direction. The time derivative is discretised by the implicit-Euler scheme and the spatial derivatives are discretised by the upwind finite difference scheme. We derive some conditions on the mesh-generating functions which are useful for the convergence of the method, uniformly with respect to the perturbation parameter. We prove that the proposed scheme on the Bakhvalov-Shishkin mesh is first-order convergent in the discrete supremum norm, which is optimal and does not require any extra computational effort compared to the standard Shishkin mesh. Numerical experiments verify the theoretical results.
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second order uniformly convergent richardson extrapolation method for singularly perturbed degenerate parabolic pdes
International Journal of Applied and Computational Mathematics, 2017Co-Authors: Anirban Majumdar, Srinivasan NatesanAbstract:This article studies the numerical solution of singularly perturbed degenerate parabolic convection–diffusion problem on a rectangular domain. The solution of this problem exhibits a boundary layer in the neighborhood of left boundary of the domain. We discretize the domain by piecewise-uniform Shishkin mesh in the spatial Direction and uniform mesh in the Temporal Direction. The numerical scheme consists of the implicit-Euler scheme for the time derivative and upwind finite difference scheme for the spatial derivatives. By applying the Richardson extrapolation technique, we improve the order of accuracy of the numerical solution from \(O (N^{-1} \ln ^2 N + \varDelta t)\) to \(O (N^{-2} \ln ^2 N + \varDelta t^2)\) (measured in the discrete supremum norm), where \(N+1\) mesh points are used in spatial Direction and \(\varDelta t\) is the step size in the Temporal Direction. Error estimates are derived. Some numerical experiments are carried out to validate the theoretical results.
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e uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations
International Journal of Computer Mathematics, 2017Co-Authors: S Gowrisankar, Srinivasan NatesanAbstract:This paper studies the numerical solutions of singularly perturbed parabolic convection–diffusion problems with a delay in time. We divide the domain using a piecewise uniform adaptive mesh in the spatial Direction and a uniform mesh in the Temporal Direction. Further, we discretize the time derivative by the backward-Euler scheme and the spatial derivatives by the upwind finite difference scheme. We obtain the maximum principle and carry out the stability analysis. Then we prove that the proposed scheme is -uniform convergence of first-order in time and first-order up to a logarithmic factor in space. Numerical results are carried out to verify the theoretical results.
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uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection diffusion problems on shishkin mesh
Applied Mathematics and Computation, 2015Co-Authors: Abhishek Das, Srinivasan NatesanAbstract:This article studies the numerical solution of singularly perturbed delay parabolic convection-diffusion initial-boundary-value problems. Since the solution of these problems exhibit regular boundary layers in the spatial variable, we use the piecewise-uniform Shishkin mesh for the discretization of the domain in the spatial Direction, and uniform mesh in the Temporal Direction. The time derivative is discretized by the implicit-Euler scheme and the spatial derivatives are discretized by the hybrid scheme. For the proposed scheme, the stability analysis is carried out, and parameter-uniform error estimates are derived. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.
Mostafa Abbaszadeh - One of the best experts on this subject based on the ideXlab platform.
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investigation of the oldroyd model as a generalized incompressible navier stokes equation via the interpolating stabilized element free galerkin technique
Applied Numerical Mathematics, 2020Co-Authors: Mostafa Abbaszadeh, Mehdi DehghanAbstract:Abstract The main objective of the current paper is to propose a meshless weak form to simulate the Oldroyd equation. At the first stage, the Crank-Nicolson method has been utilized to discreet the Temporal Direction. At the second stage, the meshless element free Galerkin technique has been employed to approximate the spatial Direction. The main equation is a non-local model as there is an integral term in the right hand side of the model. The mentioned integral term is approximated by using a difference scheme. The uniqueness and existence of the proposed numerical plan have been proved by using the Browder fixed point theorem. Furthermore, the error estimation of the developed numerical technique based on the stability and convergence order has been studied. Finally, some examples have been investigated to verify the theoretical results.
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alternating Direction implicit spectral element method adi sem for solving multi dimensional generalized modified anomalous sub diffusion equation
Computers & Mathematics With Applications, 2019Co-Authors: Mostafa Abbaszadeh, Mehdi Dehghan, Yong ZhouAbstract:Abstract The main aim of the current paper is to solve the multi-dimensional generalized modified anomalous sub-diffusion equation by using a new spectral element method. At first, the time variable has been discretized by a finite difference scheme with second-order accuracy. The stability and convergence of the time-discrete scheme have been investigated. We show that the time-discrete scheme is unconditionally stable and the convergence order is O ( τ 2 ) in the Temporal Direction. Secondly, the Galerkin spectral element method has been combined with alternating Direction implicit idea to discrete the space variable. The unconditional stability and convergence of the full-discrete scheme have been proved. By developing the proposed scheme, we need to calculate one-dimensional integrals for two-dimensional problems and two-dimensional integrals for three-dimensional problems. Thus, the used CPU time for the presented numerical procedure is lower than the two- and three-dimensional Galerkin spectral element methods. Also, the proposed method is suitable for computational domains obtained from the tensor product. Finally, two examples are analyzed to check the theoretical results.
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error analysis and numerical simulation of magnetohydrodynamics mhd equation based on the interpolating element free galerkin iefg method
Applied Numerical Mathematics, 2019Co-Authors: Mehdi Dehghan, Mostafa AbbaszadehAbstract:Abstract The MHD equation has some applications in physics and engineering. The main aim of the current paper is to propose a new numerical algorithm for solving the MHD equation. At first, the Temporal Direction has been discretized by the Crank–Nicolson scheme. Also, the unconditional stability and convergence of the time-discrete scheme have been investigated by using the energy method. Then, an improvement of element free Galerkin (EFG) i.e. the interpolating element free Galerkin method has been employed to discrete the spatial Direction. Furthermore, an error estimate is presented for the full discrete scheme based on the Crank–Nicolson scheme by using the energy method. We prove that convergence order of the numerical scheme based on the new numerical scheme is O ( τ 2 + δ m ) . In the considered method the appeared integrals are approximated using Gauss Legendre quadrature formula. Numerical examples confirm the efficiency and accuracy of the proposed scheme.
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an efficient technique based on finite difference finite element method for solution of two dimensional space multi time fractional bloch torrey equations
Applied Numerical Mathematics, 2018Co-Authors: Mehdi Dehghan, Mostafa AbbaszadehAbstract:Abstract The main aim of the current paper is to propose an efficient numerical technique for solving two-dimensional space-multi-time fractional Bloch–Torrey equations. The current research work is a generalization of [6] . The Temporal Direction is based on the Caputo fractional derivative with multi-order fractional derivative and the spatial Directions are based on the Riemann–Liouville fractional derivative. Thus, to achieve a numerical technique, the time variable is discretized using a finite difference scheme with convergence order O ( τ 2 − α ) . Also, the space variable is discretized using the finite element method. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, four test problems have been illustrated to verify the efficiency and simplicity of the proposed technique on irregular computational domains.
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a finite difference finite element technique with error estimate for space fractional tempered diffusion wave equation
Computers & Mathematics With Applications, 2018Co-Authors: Mehdi Dehghan, Mostafa AbbaszadehAbstract:Abstract An efficient numerical technique is proposed to solve one- and two-dimensional space fractional tempered fractional diffusion-wave equations. The space fractional is based on the Riemann–Liouville fractional derivative. At first, the Temporal Direction is discretized using a second-order accurate difference scheme. Then a classic Galerkin finite element is employed to obtain a full-discrete scheme. Furthermore, for the time-discrete and the full-discrete schemes error estimate has been presented to show the unconditional stability and convergence of the developed numerical method. Finally, two test problems have been illustrated to verify the efficiency and simplicity of the proposed technique.
Abhishek Das - One of the best experts on this subject based on the ideXlab platform.
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parameter uniform numerical method for singularly perturbed 2d delay parabolic convection diffusion problems on shishkin mesh
Journal of Applied Mathematics and Computing, 2019Co-Authors: Abhishek Das, Srinivasan NatesanAbstract:In this article, we study the numerical solution of a singularly perturbed 2D delay parabolic convection–diffusion problem. First, we discretize the domain with a uniform mesh in the Temporal Direction and a special mesh in the spatial Directions. The numerical scheme used to discretize the continuous problem, consists of the implicit-Euler scheme for the time derivative and the classical upwind scheme for the spatial derivatives. Stability analysis is carried out, and parameter-uniform error estimates are derived. The proposed scheme is of almost first-order (up to a logarithmic factor) in space and first-order in time. Numerical examples are carried out to verify the theoretical results.
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uniformly convergent numerical method for singularly perturbed 2d delay parabolic convection diffusion problems on bakhvalov shishkin mesh
International Journal of Mathematical Modelling and Numerical Optimisation, 2018Co-Authors: Abhishek Das, Srinivasan NatesanAbstract:In this paper, we consider a class of singularly perturbed 2D delay parabolic convection-diffusion initial-boundary-value problems. To solve this problem numerically, we use a modified Shishkin mesh (Bakhvalov-Shishkin mesh) for the discretisation of the domain in the spatial Directions and uniform mesh in the Temporal Direction. The time derivative is discretised by the implicit-Euler scheme and the spatial derivatives are discretised by the upwind finite difference scheme. We derive some conditions on the mesh-generating functions which are useful for the convergence of the method, uniformly with respect to the perturbation parameter. We prove that the proposed scheme on the Bakhvalov-Shishkin mesh is first-order convergent in the discrete supremum norm, which is optimal and does not require any extra computational effort compared to the standard Shishkin mesh. Numerical experiments verify the theoretical results.
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uniformly convergent hybrid numerical scheme for singularly perturbed delay parabolic convection diffusion problems on shishkin mesh
Applied Mathematics and Computation, 2015Co-Authors: Abhishek Das, Srinivasan NatesanAbstract:This article studies the numerical solution of singularly perturbed delay parabolic convection-diffusion initial-boundary-value problems. Since the solution of these problems exhibit regular boundary layers in the spatial variable, we use the piecewise-uniform Shishkin mesh for the discretization of the domain in the spatial Direction, and uniform mesh in the Temporal Direction. The time derivative is discretized by the implicit-Euler scheme and the spatial derivatives are discretized by the hybrid scheme. For the proposed scheme, the stability analysis is carried out, and parameter-uniform error estimates are derived. Numerical examples are presented to show the accuracy and efficiency of the proposed scheme.
Yifa Tang - One of the best experts on this subject based on the ideXlab platform.
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convergence and superconvergence of a fully discrete scheme for multi term time fractional diffusion equations
Science & Engineering Faculty, 2016Co-Authors: Yanmin Zhao, Yifa Tang, Yadong Zhang, Fawang Liu, Ian Turner, Vo AnhAbstract:Using finite element method in spatial Direction and classical L1L1 approximation in Temporal Direction, a fully-discrete scheme is established for a class of two-dimensional multi-term time fractional diffusion equations with Caputo fractional derivatives. The stability analysis of the approximate scheme is proposed. The spatial global superconvergence and Temporal convergence of order O(h2+τ2−α)O(h2+τ2−α) for the original variable in H1H1-norm is presented by means of properties of bilinear element and interpolation postprocessing technique, where hh and ττ are the step sizes in space and time, respectively. Finally, several numerical examples are implemented to evaluate the efficiency of the theoretical results.
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finite element multigrid method for multi term time fractional advection diffusion equations
International Journal of Modeling Simulation and Scientific Computing, 2015Co-Authors: Weiping Bu, Yifa Tang, Jiye YangAbstract:In this paper, a class of multi-term time fractional advection diffusion equations (MTFADEs) is considered. By finite difference method in Temporal Direction and finite element method in spatial Direction, two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained. The stability and convergence of these numerical schemes are discussed. Next, a V-cycle multigrid method is proposed to solve the resulting linear systems. The convergence of the multigrid method is investigated. Finally, some numerical examples are given for verification of our theoretical analysis.
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an implicit mls meshless method for 2 d time dependent fractional diffusion wave equation
Applied Mathematical Modelling, 2015Co-Authors: Jiye Yang, Y M Zhao, Na Liu, Yifa TangAbstract:Abstract It is well accepted that fractional partial differential equations (FPDE) can be used to model many processes for which the normal partial differential equations (PDE) fail to describe precisely. Numerical approaches seem to be promising alternatives when exact solution of FPDE is difficult to derive. However, numerical solution of FPDE encounters new challenges brought in by the fractional order derivatives. In this paper we consider the 2D time dependent fractional diffusion–wave equation (FDWE) with Caputo derivative in Temporal Direction. We discretize the fractional order derivative with finite difference method (FDM) and present a moving least squares (MLS) meshless approximation in spatial Directions which can be used to handle more complex problem domain. The convergence and stability properties of semi-discretized scheme related to time are theoretically analyzed. Finally, we conduct several numerical experiments to test our method for both regular and irregular node point distribution on rectangular and circular domain. The results indicate that the proposed method is accurate and efficient.
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two finite difference schemes for time fractional diffusion wave equation
Numerical Algorithms, 2013Co-Authors: Jianfei Huang, Yifa Tang, Luis Vazquez, Jiye YangAbstract:Time fractional diffusion-wave equations are generalizations of classical diffusion and wave equations which are used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. In this paper we construct two finite difference schemes to solve a class of initial-boundary value time fractional diffusion-wave equations based on its equivalent partial integro-differential equations. Under the weak smoothness conditions, we prove that our two schemes are convergent with first-order accuracy in Temporal Direction and second-order accuracy in spatial Direction. Numerical experiments are carried out to demonstrate the theoretical analysis.
Jiye Yang - One of the best experts on this subject based on the ideXlab platform.
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finite element multigrid method for multi term time fractional advection diffusion equations
International Journal of Modeling Simulation and Scientific Computing, 2015Co-Authors: Weiping Bu, Yifa Tang, Jiye YangAbstract:In this paper, a class of multi-term time fractional advection diffusion equations (MTFADEs) is considered. By finite difference method in Temporal Direction and finite element method in spatial Direction, two fully discrete schemes of MTFADEs with different definitions on multi-term time fractional derivative are obtained. The stability and convergence of these numerical schemes are discussed. Next, a V-cycle multigrid method is proposed to solve the resulting linear systems. The convergence of the multigrid method is investigated. Finally, some numerical examples are given for verification of our theoretical analysis.
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an implicit mls meshless method for 2 d time dependent fractional diffusion wave equation
Applied Mathematical Modelling, 2015Co-Authors: Jiye Yang, Y M Zhao, Na Liu, Yifa TangAbstract:Abstract It is well accepted that fractional partial differential equations (FPDE) can be used to model many processes for which the normal partial differential equations (PDE) fail to describe precisely. Numerical approaches seem to be promising alternatives when exact solution of FPDE is difficult to derive. However, numerical solution of FPDE encounters new challenges brought in by the fractional order derivatives. In this paper we consider the 2D time dependent fractional diffusion–wave equation (FDWE) with Caputo derivative in Temporal Direction. We discretize the fractional order derivative with finite difference method (FDM) and present a moving least squares (MLS) meshless approximation in spatial Directions which can be used to handle more complex problem domain. The convergence and stability properties of semi-discretized scheme related to time are theoretically analyzed. Finally, we conduct several numerical experiments to test our method for both regular and irregular node point distribution on rectangular and circular domain. The results indicate that the proposed method is accurate and efficient.
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two finite difference schemes for time fractional diffusion wave equation
Numerical Algorithms, 2013Co-Authors: Jianfei Huang, Yifa Tang, Luis Vazquez, Jiye YangAbstract:Time fractional diffusion-wave equations are generalizations of classical diffusion and wave equations which are used in modeling practical phenomena of diffusion and wave in fluid flow, oil strata and others. In this paper we construct two finite difference schemes to solve a class of initial-boundary value time fractional diffusion-wave equations based on its equivalent partial integro-differential equations. Under the weak smoothness conditions, we prove that our two schemes are convergent with first-order accuracy in Temporal Direction and second-order accuracy in spatial Direction. Numerical experiments are carried out to demonstrate the theoretical analysis.
