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In this paper, new group iterative numerical schemes based on the centred and rotated (skewed) seven-point finite difference discretisations are proposed for the solution of a three dimensional second order hyperbolic telegraph equation, subject to specific initial and Dirichlet boundary conditions. Both schemes are shown to be of second order accuracies and unconditionally stable. The scheme derived from the rotated grid stencil results in a reduced linear system with lower computational complexity compared to the scheme derived from the centred approximation formula. A comparative study with other common point iterative methods based on the seven-point centred difference approximation together with their computational complexity analyses is also presented. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we present the development of new explicit group relaxation methods which solve the two dimensional second order hyperbolic telegraph equation subject to specific initial and Dirichlet boundary conditions. The explicit group methods use small fixed group formulations derived from a combination of the rotated five-point finite difference approximation together with the centered five-point centered difference approximation on different grid spacings. The resulting schemes involve three levels finite difference approximations with second order accuracies. Analyses are presented to confirm the unconditional stability of the difference schemes. Numerical experimentations are also conducted to compare the new methods with some existing schemes. [Copyright &y& Elsevier]

Due to their rapid convergence properties, recent focus on iterative methods in the solution of linear system has seen a flourish on the use of gradient techniques which are primarily based on global minimisation of the residual vectors. In this paper, we conduct an experimental study to investigate the performance of several preconditioned gradient or variational techniques to solve a system arising from the so-called rotated (skewed) finite difference discretisation in the solution of elliptic partial differential equations (PDEs). The preconditioned iterative methods consist of variational accelerators, namely the steepest descent and conjugate gradient methods, applied to a special matrix 'splitting' preconditioned system. Several numerical results are presented and discussed. † E-mail: dj.evans@ntu.ac.uk [ABSTRACT FROM AUTHOR]

A second-order finite difference scheme derived from rotated discretisation formula is employed in conjunction with a preconditioner to obtain highly accurate and fast numerical solution of the two-dimensional elliptic partial differential equation. The use of a 'splitting' preconditioning strategy will be shown to improve the spectral properties of the matrix of the linear system resulting from this discretisation by minimising the eigenvalue spectrum of the transformed matrix. The application of this technique to several acceleration iterative methods, such as Simultaneous displacement, Richardson's and Chebyshev accelerated methods, are presented and discussed. † E-mail: dj.evans@ntu.ac.uk [ABSTRACT FROM AUTHOR]