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In this paper, a compact high-order implicit scheme is presented for the solution of the two-dimensional time-fractional diffusion equation. The Caputo fractional derivative and compact implicit scheme are used for the time and space derivatives respectively. The convergence of the proposed scheme will be shown to be O(τ2−β – h4), where τ, β and h are representing time step, fractional-order and space step respectively. Finally, some numerical examples are provided, which shows the accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, the formulation of explicit group iterative methods namely the fractional explicit group (FEG) and fractional explicit de-coupled group (FEDG) methods, which are based on both standard and skewed five point finite difference discretisation, are considered in solving the two dimensional second order time-fractional diffusion wave equation with damping. The Caputo formula of order α (1 < α < 2) is utilized for the fractional time derivative. The implementation of the group iterative schemes are demonstrated on solving numerical examples using Mathematica 11 software. Our numerical results of the derived group iterative schemes FEG and FEDG are more feasible and efficient as compare to its counterparts point iterative schemes. [ABSTRACT FROM AUTHOR]

In this paper, the formulation of a new fractional modified explicit group (FMEG) iterative method in solving the two-dimensional time fractional diffusion equation (TFDE) is presented. The method is formulated using the fivepoint centred difference approximation on the 2h grid stencils. The new developed method is shown to have a better convergence rate due to its lower computational complexity compared to others existing fractional group methods. Numerical experimentations of this new method, show significant improvement in execution time over the other explicit methods of the same class which enhances the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Variety of problems in science and engineering may be described by fractional partial differential equations (FPDE) in relation to space and/or time fractional derivatives. The difference between time fractional diffusion equations and standard diffusion equations lies primarily in the time derivative. Over the last few years, iterative schemes derived from the rotated finite difference approximation have been proven to work well in solving standard diffusion equations. However, its application on time fractional diffusion counterpart is still yet to be investigated. In this paper, we will present a preliminary study on the formulation and analysis of new explicit group iterative methods in solving a twodimensional time fractional diffusion equation. These methods were derived from the standard and rotated Crank- Nicolson difference approximation formula. Several numerical experiments were conducted to show the efficiency of the developed schemes in terms of CPU time and iteration number. [ABSTRACT FROM AUTHOR]

The formulation and implementation of a new explicit group method in solving the two dimensional (2D) elliptic partial differential equation (PDE) is presented. The fourth order group scheme is derived from the rotated nine-point finite difference discretization formula applied to the Poisson equation. Numerical solutions are obtained for different mesh sizes. The iterative procedure of the Fourth Order Modified Explicit Decoupled Group (FOMEDG) method is found to require lesser execution timings and iteration numbers compared to the existing Fourth Order Modified Explicit Group (FOMEG) method resulting in faster convergence rate. [ABSTRACT FROM AUTHOR]

In order to improve the accuracy of solutions of partial differential equations (PDEs), the formulation of higher order solvers may be inevitable. In this paper, we propose a new fourth-order finite difference scheme derived from the nine-point approximation formula on the rotated grid in solving the two-dimensional Poisson PDE with Dirichlet boundary conditions. Several numerical experiments are carried out and the results are compared with the analytical solutions to verify the higher accuracy of the presented schemes. [ABSTRACT FROM AUTHOR]

Improved techniques derived from the rotated finite difference operators have been developed over the last few years in solving the linear systems that arise from the discretization of various partial differential equations (PDEs). Furthermore, a higher order system can be generated from discretization of the finite difference scheme using the fourth order compact scheme generated from the second order central difference. By using compact finite differences, a new rotated point scheme with fourth-order accuracy for the two-dimensional (2D) Helmholtz equation is formed. On the other hand, the multiscale multigrid method combined with Richardson's extrapolation is first introduced by Zhang to solve the 2D Poisson equation. By combining the fourth-order rotated scheme and multiscale multigrid method with Richardson's extrapolation in the solution of the 2D Helmholtz equation, the order of accuracy of the approximation can be improved up to sixth order, and with larger mesh size, the convergence rate of these iterative methods is faster as well. Numerical experiments are conducted on the rotated scheme combined with multiscale multigrid method and Richardson's extrapolation, and the result is compared with existing point methods and multigrid method. The results show the improvements in the convergence rate and the efficiency of the newly formulated iterative scheme. [ABSTRACT FROM AUTHOR]