Abstract:
In this work, we construct novel central finite difference schemes for solving the
Helmholtz equation by following the idea of Nehrbass and Yau Shu in one and
two dimensions. We present an alternative construction by which not only central
finite difference schemes but all types of schemes (forward, backward) can be
made exact for Helmholtz equation. We give explicit expressions for the coefficient
of the central node in the standard central finite difference scheme of any order
which makes the novel scheme dispersion free in one dimension and optimal in two
dimensions. Such schemes add no cost for the implementation and one needs not to
write brand new code instead just need to change the central node coefficient with
the new one presented in this work to have highly accurate results. Comparison
of numerical solutions obtained using novel schemes with standard schemes for
Helmholtz equation in single and higher dimensions are given.
