A typical backward substitution method for the simulation of Helmholtz problems in arbitrary 2D domains

This paper presents a novel meshless method for the simulation of Helmholtz equations in arbitrary 2D domains. In the proposed method, the boundary conditions are approximated in advance to given the primary approximation of the solution. Then the final approximation is given by the summation of the primary approximation, the radial basis functions, and the related special correcting functions which are determined by the homogeneous boundary conditions. Then the approximation is substituted back to the governing equations where the unknown coefficients can be determined. The numerical examples are designed to investigate the accuracy and stability of the proposed meshless method. Furthermore, the proposed method is applied to Helmholtz problems in high-frequency regimes to show the capabilities of the method. Compared with the reference solutions, it is found that the present method is of high accuracy and rapid convergence.