Numerical Dissipation
#+CATEGORY: cfd
Numerical Dissipation :: The Unintended Convection-Diffusion of Discretization
Numerical dissipation acts like an additional diffusion term absent from the governing equations. Understanding and managing it is essential because:
In RANS, numerical dissipation adds to modeled turbulent dissipation, making Reynolds stresses ambiguous. In LES, it contaminates the subgrid-scale model that should represent only physical dissipation. In DNS, numerical dissipation must be negligible compared to physical dissipation. Across shocks, intentional numerical dissipation from limiters prevents oscillations.
Modified Equation Analysis
The modified equation technique reveals what a discrete scheme actually solves. The technique applies to the scalar advection equation. The equation is partial phi partial t plus u partial phi partial x equals zero. The scheme uses first-order upwind in space and implicit Euler in time. For simplicity, the analysis assumes steady 1D flow.
The upwind spatial discretization for steady 1D flow is:
Taylor expanding phi_{i-1} about phi_i:
Substituting:
Substituting reveals that first-order upwind introduces numerical diffusivity proportional to half u delta x, with three implications: it smears discontinuities, depends on grid resolution (refining reduces it but also changes physics), and on coarse grids can exceed physical diffusion.
Central Differencing and Numerical Dispersion vs Dissipation
Central differencing has no leading-order truncation error diffusion. Its leading error term is third-order and dispersive, not second-order and dissipative. This third-order term causes numerical dispersion, producing oscillations upstream and downstream of gradients. Numerical dissipation causes smearing. The two effects are linked. See the numerical dispersion and numerical dissipation notes. They are two sides of the same truncation error coin.
Impact on RANS, LES, and DNS
RANS simulations are relatively insulated since modeled dissipation usually dwarfs numerical dissipation on practical meshes. First-order upwind diffuses mean quantities and produces wrong profiles. LES simulations are extremely sensitive because the grid resolves energy-containing eddies and the SGS model handles only universal small scales. If grid numerical dissipation matches SGS dissipation, neither model behaves as intended—this is known as hybrid RANS/LES corruption. DNS requires delta x plus less than or equal to 1, delta y plus about 0.5, and delta z plus less than or equal to 1 in wall-bounded flow, using at least second-order central schemes to keep numerical dissipation sub-dominant.
Limiters and Shock Capture
TVD limiters introduce adaptive numerical dissipation that is minimal in smooth regions and maximal at discontinuities. The limiter function psi(r) modulates high-order reconstruction based on the smoothness indicator r from compression schemes. When psi equals 1, the scheme uses full second-order reconstruction with minimal dissipation. When psi equals 0, it uses first-order upwind with maximum dissipation. This mechanism captures shocks without spurious oscillations. The penalty is shock resolution spanning several cells. Minmod spreads shocks over 3 to 5 cells. Superbee uses 2 to 3 cells.
Quantifying Numerical Dissipation
Numerical diffusivity is not directly output by codes. Its effects are measurable through turbulent kinetic energy decay in homogeneous isotropic turbulence simulations, decay rate of passive scalar variance, thickness of resolved shocks relative to grid spacing, and comparison against analytical solutions like the Burger equation which provides exact shock structure.
RANS engineers use a qualitative check: run the same case on two grids with different cell counts. If the solution changes substantially between grids in a region where the mesh appears converged, numerical dissipation may be the culprit instead of physical modeling error.
Reducing Numerical Dissipation
Reduce numerical dissipation by using second-order or higher bounded schemes instead of first-order upwind, refining the grid, avoiding excessive under-relaxation that acts as low-pass filtering in iteration space, choosing schemes with minimal truncation error for the given stencil, and using central differencing in LES to let the SGS model carry the dissipation.
Summary
Every discretization scheme introduces numerical dissipation or dispersion via truncation error. First-order upwind introduces half u delta x effective diffusivity, which is substantial and grid-dependent. Central differencing introduces oscillations instead of smearing. In DNS, numerical dissipation must be negligible. In RANS, it is often irrelevant. In LES, it is the most dangerous error source. TVD limiters provide adaptive dissipation: on at shocks, off in smooth regions.
See Also