The artificial attenuation or smoothing of amplitude and high-frequency content introduced by a numerical algorithm (time integrator, finite difference/volume/element scheme, or filter) that mimics added viscous-like effects and reduces oscillations in the discrete solution.

Principle

Algorithms that replace or approximate differential operators, or that stabilize calculations, introduce dissipative terms (explicit or implicit) which systematically decrease certain norms or spectral components of the discrete solution; controlled dissipation trades high-frequency accuracy for robustness.

Demonstration

A first-order upwind finite-difference scheme for linear advection causes an initially sharp profile to lose amplitude and smear over time: Fourier-mode analysis shows exponential decay of high wavenumber components compared with an exact non-dissipative transport.

Misapplication

Interpreting every loss of amplitude in a simulation as physical viscous dissipation rather than as algorithmic smoothing; or deliberately adding excessive artificial viscosity to mask a stability problem instead of addressing the discretization error.

Consequence

When recognized and controlled, numerical dissipation stabilizes computations, damps nonphysical oscillations, and can permit larger time steps; excessive dissipation, however, destroys important small-scale features and biases quantitative results.

Reversal

Numerical dispersion: errors that primarily alter phase and wave speeds rather than amplitude, producing spurious oscillatory tails and ringing without systematic decay of mode amplitudes.

Boundary

Applies to algorithmic effects in discrete models (spatial discretizations, time integrators, numerical filters); excludes physical viscosity, model-form damping terms that are part of the governing PDE, and statistical smoothing applied as postprocessing.

Semantic Tension

Often contrasted with numerical dispersion — both are numerical artifacts from approximation but act differently on amplitude versus phase; practitioners must distinguish physical dissipation from algorithmic dissipation when validating models.

Synthesis

Numerical dissipation is the deliberate or incidental damping of discrete solution amplitude introduced by numerical methods; it is governed by the approximation and stabilization mechanisms of the scheme, useful for robustness but harmful for unresolved small-scale accuracy.