  In this work, a new compact sixth order accurate finite difference scheme for
the two and three-dimensional Helmholtz equation is presented. The main
significance of the proposed scheme is that its sixth order leading truncation
error term does not explicitly depend on the associated wave number. This makes
the scheme robust to work for the Helmholtz equation even with large wave
numbers. The convergence analysis of the new scheme is given. Numerical results
for various benchmark test problems are given to support the theoretical
estimates. These numerical results confirm the accuracy and robustness of the
proposed scheme.

Numerical Analysis