Numerical Issues

While speed increases brought by faster computer goes significantly to what can be accomplished using DNS, smarter programming, faster algorithms and novel theoretical tools receive continued emphasis to make progress in DNS.

For periodically assumed flows, fast Fourier series methods have enabled numerous temporal DNS studies. For the spatial DNS approach, high order ($\ge 4$th) finite difference methods are commonly used in DNS codes. Due to the advances made by Lele [8], high-order compact difference techniques have been included in more recent DNS efforts. Spectral element and collocation methods have been used for spatial discretization around complex geometries [2]. Chebyshev collocation techniques (which use polynomials instead of trigonometric terms and are hence for non-periodic flows) have been used in boundary-layer and channel flow problems. Also, numerous DNS studies have used schemes like Adam-Bashforth, Runge-Kutta, Crank-Nicolson, and time-splitting approaches for time advancement.

Often Poisson or Helmholtz equations (Dirichlet and Neumann boundary conditions) must be solved during the course of a DNS. A Gauss-Siedel like iteration procedure and direct solvers have been used in DNS codes for these. Fast serial and parallel high-order direct solvers for Poisson and Helmholtz equations have been tested for speed and accuracy.