Numerical integration of multi-dimensional highly oscillatory integrals, based on eRPIM

Abstract

This paper discusses the computation of multi-dimensional highly oscillatory integrals. The approach here denoted by "L-eRPIM" adopts the Levin collocation method and the enriched type radial point interpolation method (eRPIM) appropriately, showing improvement in accuracy as the frequency increases. In this approach a multi-dimensional highly oscillatory integral is first converted into a partial differential equation (PDE) by Levin collocation method. Then by eRPIM the solution of the resulting PDE is obtained to compute the value of integral. As the advantages of this approach we can mention that the well known results concerning the order of error in terms of frequency ? and dimension d are still valid and sensitivity to the shape parameter of the RBFs can also be decreased by adding more monomial and trigonometric basis functions. Numerical and practical examples are presented to demonstrate the efficiency and accuracy of the proposed method.