Fast and stable augmented Levin methods for highly oscillatory and singular integrals

Abstract

In this paper, augmented Levin methods are proposed for the computation of oscillatory integrals with stationary points and an algebraically or logarithmically singular kernel. Different from the conventional Levin method, to overcome the difficulties caused by singular and stationary points, the original Levin ordinary differential equation (Levin-ODE) is converted into an augmented ODE system, which can be fast and stably implemented with a cost of O ( n log ⁡ n O(n\log n ) by applying sparse and fast spectral methods together with the truncated singular value decomposition. The established asymptotics and convergence show that these schemes become more accurate as the frequency increases and are super-algebraically convergent. The effectiveness and accuracy were tested by numerical examples, showing perfect coincidence with the estimates.

Numerical Integration of Highly Oscillatory Functions with and without Stationary Points

“…The given examples show that the deterioration of the properties of the phase function leads to the need to increase the number of collocation points to ensure the required accuracy of the algorithm. To overcome the difficulties that arise and reduce the accuracy, the partitioning the integration interval into subintervals "without singularities" and "without stationary points inside" are often used [19,39,42]. The use of a stable operation of multiplying a tridiagonal integration matrix by a vector of interpolation coefficients allows for a significant increase in the number of collocation points without compromising the accuracy of the approximation of the anti-derivative.…”

“…In this case, the algorithms considered above do not work reliably enough, since they make it necessary to solve ill-conditioned SLAEs. In this regard, we note the recently published work devoted to an in-depth study of Levin's method [19].…”

“…In [11,19], various options for implementing this method in the case of various nonlinear frequency functions of a fairly general form were studied, and many applied problems were solved. The method of integrating functions with a linear phase was studied in [20].…”