A new shifted Laplacian preconditioner for solving the Helmholtz wave equation

The shifted Laplacian operator, combined with multigrid methods, is widely utilized as a preconditioner for solving the 3D Helmholtz equation. However, its effectiveness diminishes as the solving frequency increases. To enhance the performance of the shifted Laplacian operator, we propose that the efficacy of the preconditioner is related to the percentage of seismic wave energy absorbed by a single grid. Following this premise, we derived a new shifted Laplacian preconditioner base on theoretical solutions from wave equation theory. Subsequently, we employed a V-cycle multigrid method integrated with the new shifted Laplacian preconditioner and used the generalized minimal residual method as the smoothing operator to solve the 3D Helmholtz equation. To verify the effectiveness of the new preconditioner, the numerical solutions of the wave equation were compared with the analytical solutions, which confirmed the iterative method’s efficacy for solving the 3D Helmholtz equation. Additionally, we conducted experiments using three common 3D models. The numerical results indicate that, at low frequencies, the optimized shifted Laplacian operator achieves similar computation times to the original operator. However, at medium to high frequencies, the new shifted Laplacian operator significantly reduces computation time, thus confirming its enhanced performance in solving the 3D Helmholtz equation.