A fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity

Abstract

In this paper, we use orthogonal spline collocation methods (OSCM) for the one dimensional Helmholtz equation with discontinuous coefficients. We discuss the existence uniqueness results and establish optimal error estimates. We use piecewise Hermite cubic basis functions to approximate the solution. Finally, we perform some numerical experiments and validate the theoretical results. Comparative to existing methods, we prove that the orthogonal spline collocation methods (OSCM) handles the discontinuous coefficients effectively and gives optimal order of convergence for $$\Vert y-y_h\Vert _{L^{\infty }}$$ -norm and superconvergent result for $$\Vert y'-y'_h\Vert _{L^{\infty }}$$ -norm at the grid points.