A New Class of Highly Accurate Differentiation Schemes Based on the Prolate Spheroidal Wave Functions

Abstract

We introduce a new class of numerical differentiation schemes constructed for the efficient solution of time-dependent PDEs that arise in wave phenomena. The schemes are constructed via the prolate spheroidal wave functions (PSWFs). Compared to existing differentiation schemes based on orthogonal polynomials, the new class of differentiation schemes requires fewer points per wavelength to achieve the same accuracy when it is used to approximate derivatives of bandlimited functions. In addition, the resulting differentiation matrices have spectral radii that grow asymptotically as m for the case of first derivatives, and m 2 for second derivatives, with m being the dimensions of the matrices. The above results mean that the new class of differentiation schemes is more efficient in the solution of time-dependent PDEs compared to existing schemes such as the Chebyshev collocation method. The improvements are particularly prominent in large-scale time-dependent PDEs, in which the solutions contain large numbers of wavelengths in the computational domains.

On hp-convergence of prolate spheroidal wave functions and a new well-conditioned prolate-collocation scheme

“…The PSWFs provide an optimal tool in approximating general bandlimited functions (see e.g., [26,25,32,15]). On the other hand, being the eigenfunctions of a singular Sturm-Liouville problem (cf.…”

“…With this in mind, we first introduce the Kong-Rokhlin's rule in [15] for pairing up (c, N ) that guarantees high accuracy in integration and differentiation of bandlimited functions, but it requires computing λ N . In this section, we first propose a practical mean for its implementation.…”

“…In practice, a quite safe choice is c = N/2 (see e.g., [7,27]). From a different perspective, Kong and Rokhlin [15] proposed a useful rule for pairing up (c, N ). The starting point is a prolate quadrature rule, say (2.17).…”

“…• We propose a practical approximation to Kong-Rokhlin's rule for pairing up (c, N ) (see [15]), and demonstrate that the collocation scheme using this rule significantly outperforms the Legendre polynomial-based method when the involved solution is bandlimited. For example, the portion of discrete eigenvalues of the prolate differentiation matrix that approximates the eigenvalues of the continuous operator to 12-digit accuracy is about 87% against 25% for the Legendre case (see Subsection 3.2).…”

Ball prolate spheroidal wave functions in arbitrary dimensions

“…The properties inherent to these functions have subsequently attracted many attentions for decades. Within the last few years, there has been a growing research interest in various aspects of the PSWFs including analytic and asymptotic studies [48,12,33,9], approximation with PSWFs [34,8,49,47,31], numerical evaluations [10,13,42,18,21,3,28], development of numerical methods using this bandlimited basis [14,24,45,20]. In particular, we refer to the monographs [19,32] and the recent review paper [43] for many references therein.…”