Effectiveness of Floating-Point Precision on the Numerical Approximation by Spectral Methods

With the fast advances in computational sciences, there is a need for more accurate computations, especially in large-scale solutions of differential problems and long-term simulations. Amid the many numerical approaches to solving differential problems, including both local and global methods, spectral methods can offer greater accuracy. The downside is that spectral methods often require high-order polynomial approximations, which brings numerical instability issues to the problem resolution. In particular, large condition numbers associated with the large operational matrices, prevent stable algorithms from working within machine precision. Software-based solutions that implement arbitrary precision arithmetic are available and should be explored to obtain higher accuracy when needed, even with the higher computing time cost associated. In this work, experimental results on the computation of approximate solutions of differential problems via spectral methods are detailed with recourse to quadruple precision arithmetic. Variable precision arithmetic was used in Tau Toolbox, a mathematical software package to solve integro-differential problems via the spectral Tau method.