A highly accurate Helmholtz scheme for modeling scattering wave Original Research Article

We present in this paper a compact scheme for modelling underwater wave propagation in large-coastal areas. The proposed Helmholtz scheme relates the derivative terms uxxxx and uxx to u itself at four adjacent nodal points to yield a five-point stencil implicit scheme with eighth-order accuracy. Besides enjoying high-order accuracy and good stability properties, the proposed scheme can provide oscillation-free solutions provided that the grid size is appropriately chosen. The underlying theory is the M-matrix theory. Computations have been carried out for several problems which are amenable to exact solutions. Agreement with the exact solutions is generally good even in a grid system involving fewer mesh points, thus demonstrating the efficiency of the proposed scheme implemented within the alternating direction context. The compact scheme has been applied to problems bounded partly by a scatter surface. Both incident and scattering wave propagation problems have been studied with great success.