The theme of this book is high-accuracy FD methods. Only for pseudospectral methods (Chapter 2) was the accuracy increased to its infinite order limit. Some compromises between order of accuracy and other features (such as handling of boundaries, numerical stability, etc.) are usually needed. This appendix focuses on cases where high orders of accuracy are readily achievable, but some of these orders can favorably be exchanged for the introduction of free parameters which then can be utilized for alternative purposes. Examples include increasing the wave number range for an FD scheme, enhanced Gregory quadrature, and accelerating ODE solvers for wave equations when using multicore processors.

Gaussian quadrature can be very effective on smooth data that is available at highly specific node locations. A more common situation is that data is equispaced (if not created explicitly for the purpose of such quadrature). The most effective quadrature methods that are then available relate closely to FD approximations. A particularly noteworthy method was introduced by Gregory already in 1670 (predating the descriptions of calculus by Leibniz and Newton). In cases when the function to be integrated happens to be analytic, complex plane FD approximations can be used for highly accurate contour integration. Given the close relation between integrals and equispaced sums, FD-based methods can be very effective also for infinite sums.

Although the history of fractional order derivatives is nearly as long as that of regular (integer order) derivatives, their range of applications has increased dramatically in recent decades. In contrast to an integer order derivative, a fractional order derivative is not a local operator. It is most often expressed as an integral between some “base point” and an “evaluation point.” Although this integral is singular at least at the evaluation point, the FD perspectives provided in the previous chapters can still be applied and have recently led to a very high-order accurate method for their approximation. This is the case both for purely real-valued functions and for analytic functions in the complex plane. Fractional order Laplace operators are also frequently encountered in various applications, and methods for their approximation are discussed.