# Interdisciplinary Applied Mathematics

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dQn+1

dn

At

du^_

dt

and therefore the magnitude of the boundary-divergence flux is controlled directly by the time step. We can reduce the boundary-divergence errors further by using a multistep approximation to represent the right-hand side in equation (14.4):

dpn+1

dn

Jp-1

-vn ■    (VХш)п-q,

where Jp is the number of previous steps from which information is used. Following the above argument we find that

ж (At)Jp,

dQn+1

dn

and therefore the boundary-divergence flux can be made arbitrarily small by controlling the time step At. Note that for the inviscid pressure boundary condition

dpn+1 dn _°’

the boundary-divergence flux is 0(1), independent of the size of the time step At.

To relate the boundary-divergence to the overall accuracy of the velocity field we consider the equation that the divergence Q = Qn+1 satisfies, i.e.,

Q_

At

Yov V2Q

0,

where we set the right-hand side to zero, since the pressure satisfies a consistent Poisson equation, and the divergence at previous time steps (Qn, Qn-1, ■■■) is assumed zero; 70 is a coefficient due to time-discretization (Karniadakis et al., 1991). It is clear that there exists a numerical boundary layer of thickness 8 = л/70ITAt, so that Q = Qwe~s/S, and thus the boundary-divergence is Qw = -8(dQ/dn)w. (Here s is a general coordinate normal to the boundary). Similarly, from order of magnitude analysis we have Qw = 0(dv/dn), and therefore