Complete centered finite difference method for Helmholtz equation

Abstract

Abstract A new approach in the finite difference framework is developed, which consists of three steps: choosing the dimension of the local approximation subspace, constructing a vector basis for this subspace, and determining the coefficients of the linear combination. New schemes were developed to form the basis of the local approximation subspace, which were derived by approximating only the k 2 u term of the Helmholtz equation. The construction of a basis of the local approximation subspace allows the new approach to be able to represent any finite difference scheme that belongs to this subspace. The new method is both consistent and capable of minimizing the dispersion relation for all stencils in all dimensions. In the one-dimensional case and 3-point stencil, pollution error is eliminated. In the two-dimensional (2D) case and 5-point stencil, the Complete Centered Finite Difference Method presents a dispersion relation equivalent to Galerkin/Least-Squares Finite Element Method. In the 2D case and 9-point stencil, two versions were developed using two different bases for the local approximation space. Both versions are equivalent and exhibit a dispersion relation similar to Quasi Stabilized Finite Element Method. Additionally, the dispersion analysis revealed a connection between the coefficients of the linear system and the stencil symmetry.

Keywords

• dispersion analysis
• finite difference method
• Helmholtz equation
• local truncation error
• stabilization