The discretization of the mesh has an undesirable effect on the calculated material dispersion if the elements are too large or their longitudinal extension is transverse to the direction of flow. The following figure shows how the plume spreads much more in areas with unfavourable discretisation than with finer discretisation. This effect is known as numerical dispersion.

Influence of discretization on the results of a mass transfer calculation

(from: König, C.: Numerical computation of the three-dimensional mass flow in the groundwater. – Technisch-wissenschaftliche Mitteilungen des Instituts für konstruktiven Ingenieurbau, Ruhr-Universität Bochum, 91-93, (Bochum))

Taking the following criteria into account, this effect can be kept so small that it does not exceed the natural dispersion:

…and

L is the longest extension of an element in the direction of flow, 𝛼L is the dispersion coefficient in the longitudinal direction of flow.

Oscillations of the numerical solution are related to the fact that the transport equation is a differential equation with a partially hyperbolic character, which causes difficulties since the algorithms used are primarily designed for solving differential equations of parabolic type.

If the advective component of the transport is large relative to the dispersive component, the hyperbolic character of the equation predominates.

A measure of this ratio is the Peclet number P_e, which should be less than 2 if possible so that the hyperbolic component does not predominate too strongly

Dependence between Peclet number and oscillations (after König, 1991)

The Peclet number describes the ratio of the advection component to the dispersion component (D) in relation to a characteristic length (edge length of the elements, Δ𝑙).

The lower the Peclet numbers become, the fewer iterations are required to reach predefined maximum value of the residuals. As soon as this dimensionless number exceeds the value 10, it is no longer guaranteed that the solution will converge. An optimal local discretisation results in a Peclet number smaller than 2.

Time discretization