  We present a new higher-order accurate finite difference explicit jump
Immersed Interface Method (HEJIIM) for solving two-dimensional elliptic
problems with singular source and discontinuous coefficients in the irregular
region on a compact Cartesian mesh. We propose a new strategy for discretizing
the solution at irregular points on a nine point compact stencil such that the
higher-order compactness is maintained throughout the whole computational
domain. The scheme is employed to solve four problems embedded with circular
and star shaped interfaces in a rectangular region having analytical solutions
and varied discontinuities across the interface in source and the coefficient
terms. We also simulate a plethora of fluid flow problems past bluff bodies in
complex flow situations, which are governed by the Navier-Stokes equations;
they include problems involving multiple bodies immersed in the flow as well.
In the process, we show the superiority of the proposed strategy over the EJIIM
and other existing IIM methods by establishing the rate of convergence and grid
independence of the computed solutions. In all the cases our computed results
extremely close to the available numerical and experimental results.
