Harmonic-Enriched Reproducing Kernel Approximation for Highly Oscillatory Differential Equations

Abstract

The harmonic-enriched reproducing kernel (HRK) approximation together with collocation method is introduced to circumvent the discretization restriction for highly oscillatory partial differential equations (PDEs). It is first shown that to embed the harmonic function with a desired frequency in the HRK, both sine and cosine with the same frequency should be included in the basis vector for construction of HRK approximation. The HRK and its implicit derivatives are then used in the collocation method to effectively obtain solutions of oscillatory PDEs. The standard monomials can be included together with harmonic functions in the HRK and the reproducing conditions can be exactly satisfied with a complete set of basis functions. For PDEs with semi-harmonic solutions, the present method yields more accurate results compared with the standard reproducing kernel (RK) when a coarse discretization is used. On the other hand, when the discretization is refined, the HRK exhibits a similar convergence behavior as the standard RK. The effectiveness of the present method is demonstrated using highly oscillatory 2nd order and 4th order PDEs. The accuracy and performance of this method are compared with standard RK with the collocation method and the finite element method (FEM).