A Fast Augmented Immersed Interface Method for Solving Helmholtz Equation With Interface in Cylindrical Coordinates

This paper presents a fast algorithm for solving the axisymmetric Helmholtz equation with interfaces in cylindrical coordinates using an augmented immersed interface method. Away from the interface, a traditional second-order finite difference scheme is used; near the interface, correction terms containing augmented variables are added to the finite difference scheme to achieve global second-order convergence. The augmented variables, which are unknowns, are incorporated into a Schur complement system that is solved using the generalized minimal residual iterative method. Once the augmented variables have been determined, an existing cyclic reduction algorithm is employed to rapidly solve the discretized system. This approach not only enhances computational efficiency for solving axisymmetric Helmholtz equations with complex interfaces or boundaries but also effectively addresses high wave number problems and obstacle problems. Finally, several numerical examples with different boundary conditions are provided to illustrate the effectiveness and reliability of the proposed method.