A new class of highly accurate differentiation schemes based on the prolate spheroidal wave functions

We introduce a new class of numerical differentiation schemes 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 m2 for second derivatives, with m being the dimensions of the matrices. The results mean that the new class of differentiation schemes is more efficient in the solution of time-dependent PDEs involving bandlimited functions when compared to existing schemes such as the Chebyshev collocation method. The improvements are particularly prominent in large-scale time-dependent PDEs whose solutions contain large numbers of wavelengths in the computational domains.