# Development of a new sixth order accurate compact scheme for two and three dimensional Helmholtz equation

## TL;DR

This paper introduces a sixth order compact finite difference scheme for the 2D and 3D Helmholtz equation that remains accurate and robust even at large wave numbers, supported by theoretical analysis and numerical tests.

## Contribution

A novel sixth order compact scheme for the Helmholtz equation with wave number independent leading error term, enhancing robustness for high-frequency problems.

## Key findings

- Scheme achieves sixth order accuracy in 2D and 3D
- Numerical tests confirm robustness at large wave numbers
- Theoretical convergence analysis supports numerical results

## Abstract

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.