Numerical Simulations of Complex Helmholtz Equations Using Two-Block Splitting Iterative Schemes with Optimal Values of Parameters

Abstract

For a two-block splitting iterative scheme to solve the complex linear equations system resulting from the complex Helmholtz equation, the iterative form using descent vector and residual vector is formulated. We propose splitting iterative schemes by considering the perpendicular property of consecutive residual vector. The two-block splitting iterative schemes are proven to have absolute convergence, and the residual is minimized at each iteration step. Single and double parameters in the two-block splitting iterative schemes are derived explicitly utilizing the orthogonality condition or the minimality conditions. Some simulations of complex Helmholtz equations are performed to exhibit the performance of the proposed two-block iterative schemes endowed with optimal values of parameters. The primary novelty and major contribution of this paper lies in using the orthogonality condition of residual vectors to optimize the iterative process. The proposed method might fill a gap in the current literature, where existing iterative methods either lack explicit parameter optimization or struggle with high wave numbers and large damping constants in the complex Helmholtz equation. The two-block splitting iterative scheme provides an efficient and convergent solution, even in challenging cases.