Fast integration of rapidly oscillatory functions
Journal of Computational and Applied Mathematics
A comparison of some methods for the evaluation of highly oscillatory integrals
Journal of Computational and Applied Mathematics - Numerical evaluation of integrals
Think globally, act locally: solving highly-oscillatory ordinary differential equations
Applied Numerical Mathematics
Extended quadrature rules for oscillatory integrands
Applied Numerical Mathematics
On the Evaluation of Highly Oscillatory Integrals by Analytic Continuation
SIAM Journal on Numerical Analysis
RCMS: right correction Magnus series approach for oscillatory ODEs
Journal of Computational and Applied Mathematics
Truncation Errors in Exponential Fitting for Oscillatory Problems
SIAM Journal on Numerical Analysis
Is Gauss Quadrature Better than Clenshaw-Curtis?
SIAM Review
Complex Gaussian quadrature of oscillatory integrals
Numerische Mathematik
Exponentially fitted quadrature rules of Gauss type for oscillatory integrands
Applied Numerical Mathematics
On the convergence of Filon quadrature
Journal of Computational and Applied Mathematics
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In this paper we revisit some quadrature methods for highly oscillatory integrals of the form $\int_{-1}^{1}f(x)e^{\mathrm{i}\omega x}dx,\omega0$ . Exponentially Fitted (EF) rules depend on frequency dependent nodes which start off at the Gauss-Legendre nodes when the frequency is zero and end up at the endpoints of the integral when the frequency tends to infinity. This makes the rules well suited for small as well as for large frequencies. However, the computation of the EF nodes is expensive due to iteration and ill-conditioning. This issue can be resolved by making the connection with Filon-type rules. By introducing some S-shaped functions, we show how Gauss-type rules with frequency dependent nodes can be constructed, which have an optimal asymptotic rate of decay of the error with increasing frequency and which are effective also for small or moderate frequencies. These frequency-dependent nodes can also be included into Filon-Clenshaw-Curtis rules to form a class of methods which is particularly well suited to be implemented in an automatic software package.