Spectrally accurate numerical quadrature formulas for a class of periodic Hadamard Finite Part integrals by regularization.

We consider the numerical computation of Hadamard Finite Part (HFP) integrals K m (t ; u) = ⨎ 0 T S m (π (x − t) T) u (x) d x , 0 < t < T , m ∈ { 1 , 2 , ... } , where u (x) is T -periodic and sufficiently differentiable and S 2 r − 1 (y) = cos ⁡ y sin 2 r − 1 ⁡ y , S 2 r (y) = 1 sin 2 r ⁡ y , r = 1 , 2 , 3 , .... For each m , we regularize the HFP integral K m (t ; u) and show that K m (t ; u) = K 0 (t ; U m) ≡ ∫ 0 T (log ⁡ | sin ⁡ π (x − t) T |) U m (x) d x , U m (x) being some linear combination of the first m derivatives of u (x). We then propose to approximate K m (t ; u) by the quadrature formula Q m , n (t ; u) ≡ K m (t ; ϕ n) , where ϕ n (x) is the n th -order balanced trigonometric polynomial that interpolates u (x) on [ 0 , T ] at the 2 n equidistant points x n , k = k T 2 n , k = 0 , 1 , ... , 2 n − 1. The implementation of Q m , n (t ; u) is simple, the only input needed for this being the 2 n function values u (x n , k) , k = 0 , 1 , ... , 2 n − 1. Using Fourier analysis techniques, we develop a complete convergence theory for Q m , n (t ; u) as n → ∞ and prove that it enjoys spectral convergence when u ∈ C ∞ (R). We illustrate the effectiveness of Q m , n (t ; u) with numerical examples for m = 0 , 1 , ... , 5. We also show that the HFP integral ⨎ 0 T f (x , t) d x of any T -periodic integrand f (x , t) that has m th order poles at x = t + k T , k = 0 , ± 1 , ± 2 , ... , but is sufficiently differentiable in x on R ∖ { t ± k T } k = 0 ∞ , can be expressed in terms of the K s (t ; u (⋅ , t)) , where u (x , t) is a T -periodic and sufficiently differentiable function in x on R that can be computed from f (x , t). Therefore, ⨎ 0 T f (x , t) d x can be computed efficiently using our new numerical quadrature formulas Q s , n (t ; u (⋅ , t)) on the individual K s (t ; u (⋅ , t)). Again, only 2 n function evaluations, namely, u (x n , k , t) , k = 0 , 1 , ... , 2 n − 1 , are needed for the whole process. [ABSTRACT FROM AUTHOR]