A Well-Conditioned Spectral Galerkin–Levin Method for Highly Oscillatory Integrals

This paper addresses the numerical evaluation of highly oscillatory integrals by developing a spectral Galerkin–Levin approach that efficiently solves Levin’s differential equation formulation for such integrals. The method employs Legendre polynomials as basis functions to approximate the solution, leveraging their orthogonality and favorable numerical properties. A key finding is that the Galerkin–Levin formulation is invariant with respect to the choice of polynomial basis—be it monomials or classical orthogonal polynomials—although the use of Legendre polynomials leads to a more straightforward derivation of practical quadrature rules. Building on this, this paper derives a simple and efficient numerical quadrature for both scalar and matrix-valued highly oscillatory integrals. The proposed approach is computationally stable and well-conditioned, overcoming the limitations of collocation-based methods. Several numerical examples validate the method’s high accuracy, stability, and computational efficiency.