Sixth-order quasi-compact difference schemes for 2D and 3D Helmholtz equations

Highlights：

• What Is New of This PaperIn this paper, sixth-order QCD schemes are proposed to solve the boundary value problems of multi-dimensional Helmholtz equations with variable wavenumber. To the best of our knowledge, there were no any explicit sixth-order compact difference schemes without containing the derivatives of given functions whose global convergence order can be theoretically guaranteed in the meantime.

• Significance of This PaperIn this paper, sixth-order QCD schemes are constructed to solve the boundary value problems of two-dimensional and three-dimensional Helmholtz equations with variable wavenumber. Our strategy is to discretize the equation by the fourth-order compact scheme at the improper interior grid points that adjoin the boundary, while the sixth-order scheme, where it is compact only for the unknowns, is exploited to the proper interior grid points that are without adjoining the boundary. Different from the classic sixth-order schemes in the literature, this treatment makes the proposed schemes only require the values of source term and wavenumber on the grid points instead of the derivatives of given functions. Theoretically, we prove the QCD schemes can achieve global sixth-order accuracy for non-positive constant wavenumber. Finally, numerical experiments are carried out to manifest the accuracy and convergence order of proposed numerical methods for general wavenumber.

• HighlightsSixth-order QCD schemes are constructed to solve multi-dimensional Helmholtz equations with variable wavenumber. The proposed schemes do not contain the derivatives of given functions and without involving the additional calculation of the second boundary layer for unknowns. Theoretically, we prove the QCD schemes can achieve the global sixth-order accuracy.