Spectral treatment of the Navier–Stokes nonlinear terms in the presence of coordinate singularities

Mathematical physics equations in polar and cylindrical coordinates comprise coefficients that are singular at $r=0$. Singularity along polar axis may diminish accuracy and computational efficiency of classical spectral methods, particularly for nonlinear problems. We present fast and accurate spectral method which avoids the singularity problem and is suitable for calculation of quadratic nonlinearities of differential equations in physics. Algorithm is explained by giving an example of the nonlinear terms treatment in three-dimensional incompressible Navier–Stokes equations describing turbulent flows in a periodic circular pipe. Our approach partly utilizes the special behaviour of analytic functions near the singularity and makes use of Chebyshev parity-restricted polynomial basis for radial expansions. New algorithm may have wider applicability including problems with cylindrical symmetry in situations when computational domain contains coordinate singularities.