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

Abstract

We consider the numerical computation of Hadamard Finite Part (HFP) integralsKm(t;u)=⨎0TSm(π(x−t)T)u(x)dx,0<t<T,m∈{1,2,…}, where u(x) is T-periodic and sufficiently differentiable andS2r−1(y)=cos⁡ysin2r−1⁡y,S2r(y)=1sin2r⁡y,r=1,2,3,…. For each m, we regularize the HFP integral Km(t;u) and show thatKm(t;u)=K0(t;Um)≡∫0T(log⁡|sin⁡π(x−t)T|)Um(x)dx,Um(x) being some linear combination of the first m derivatives of u(x). We then propose to approximate Km(t;u) by the quadrature formula Qm,n(t;u)≡Km(t;ϕn), where ϕn(x) is the nth-order balanced trigonometric polynomial that interpolates u(x) on [0,T] at the 2n equidistant points xn,k=kT2n, k=0,1,…,2n−1. The implementation of Qm,n(t;u) is simple, the only input needed for this being the 2n function values u(xn,k), k=0,1,…,2n−1. Using Fourier analysis techniques, we develop a complete convergence theory for Qm,n(t;u) as n→∞ and prove that it enjoys spectral convergence when u∈C∞(R). We illustrate the effectiveness of Qm,n(t;u) with numerical examples for m=0,1,…,5.We also show that the HFP integral ⨎0Tf(x,t)dx of any T-periodic integrand f(x,t) that has mth order poles at x=t+kT, k=0,±1,±2,…, but is sufficiently differentiable in x on R∖{t±kT}k=0∞, can be expressed in terms of the Ks(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, ⨎0Tf(x,t)dx can be computed efficiently using our new numerical quadrature formulas Qs,n(t;u(⋅,t)) on the individual Ks(t;u(⋅,t)). Again, only 2n function evaluations, namely, u(xn,k,t), k=0,1,…,2n−1, are needed for the whole process.