Abstract: In this paper, the formulation of a new group explicit
method with a fourth order accuracy is described in solving the two
dimensional Helmholtz equation. The formulation is based on the
nine-point fourth order compact finite difference approximation
formula. The complexity analysis of the developed scheme is also
presented. Several numerical experiments were conducted to test the
feasibility of the developed scheme. Comparisons with other existing
schemes will be reported and discussed. Preliminary results indicate
that this method is a viable alternative high accuracy solver to the
Helmholtz equation.
Abstract: Several numerical schemes utilizing central difference
approximations have been developed to solve the Goursat problem.
However, in a recent years compact discretization methods which
leads to high-order finite difference schemes have been used since it
is capable of achieving better accuracy as well as preserving certain
features of the equation e.g. linearity. The basic idea of the new
scheme is to find the compact approximations to the derivative terms
by differentiating centrally the governing equations. Our primary
interest is to study the performance of the new scheme when applied
to two Goursat partial differential equations against the traditional
finite difference scheme.