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Direct numerical simulation of a turbulent fluid numerically solves the specified governing equations while resolving the dynamically relevant turbulent and dissipative scales. It does not replace unresolved turbulent motions with a subgrid-scale model. The calculation still assumes a continuum model, boundary conditions and a finite numerical representation; direct does not mean exact or atomistic.

A published example solves the coupled velocity, thermal and compositional equations for fingering convection with a pseudospectral method and triply periodic perturbations. The authors explicitly require resolving the dissipation scale because they use no subgrid-scale model, and inspect composition and vorticity fields to assess resolution. The simulations inform a reduced transport prescription for stellar calculations, rather than resolving an entire star.^[ref]

An exact laminar check helps explain numerical verification. For unforced incompressible Navier-Stokes equations with constant kinematic viscosity $\nu$, take constant pressure andin a periodic box containing whole wavelengths. The divergence vanishes and $u\cdot \nabla u=0$, so the equation reduces to ${\partial }_{t}u=\nu {\partial }_y^{2}u$. The mean kinetic energy per unit mass is thereforeA solver should recover this decay and energy budget under numerical refinement. Passing the check is useful but does not establish that a turbulent calculation resolves its smallest relevant scales. Direct numerical simulation also needs adequate box size, time resolution and statistical sampling.

A pseudospectral method represents an approximate solution through a global basis or its interpolating values and imposes equations at selected collocation points. In a periodic problem the basis is often a Fourier series; on an interval, polynomial collocation is common. The adjective spectral refers to the representation, not to computing the eigenvalues of an arbitrary matrix.^[ref]

For periodic data at $x_j=2\pi j/N$, the discrete Fourier transform gives coefficients ${\hat{u}}_k$ of a trigonometric interpolant. Its derivative is obtained by multiplying coefficient $k$ by $ik$ and transforming back. For nonlinear equations, products can be evaluated on the physical grid, while derivatives are computed in Fourier coordinates. A Fast Fourier transform makes these representation changes efficient.

The grid can confuse unresolved frequencies with resolved ones. With $N=8$, take $u(x)=\cos (3x)$. Its exact square isAt the eight grid points, $\cos (6x_j)=\cos (2x_j)$, so the sampled product appears to contain wave number $2$ instead of $6$. This is aliasing created by the nonlinear product, even though the input wave number $3$ was representable. Additional grid resolution, padding or appropriate filtering is needed when such products are used to evolve retained modes.

For sufficiently smooth solutions, spectral representations can converge rapidly as resolution increases. Discontinuities, boundary incompatibilities and unresolved fine structure change that behavior. A pseudospectral method therefore needs checks on both approximation error and the nonlinear operations performed on its grid.