A higher-order orthogonal collocation technique for discontinuous two-dimensional problems with Neumann boundary conditions

In this paper, the orthogonal spline collocation method (OSCM) is employed to address the solution of the Helmholtz equation in two-dimensional problems. It is characterized by discontinuous coefficients with certain wave numbers. The solution is approximated by employing distinct basis functions, namely, monomial along the x-direction and Hermite along the y-direction. Additionally, to solve the two-dimensional problems efficiently in the sense of computational cost with fewer operation counts, the matrix decomposition algorithm (MDA) is used to convert them into a set of one-dimensional problems. As a consequence, the resulting reduced matrix becomes non-singular in discrete cases. To assess the performance of the proposed numerical scheme, a grid refinement analysis is conducted to incorporate various wave coefficients of the Helmholtz equation. The illustrations and examples demonstrate a higher order of convergence compared to existing methods.