3.5 Linear Equations with Variable Coefficients
155
and that the spatial domain 'D is pcriodie or there is no flow through to the boundary of D. Under these assumptions
= 2 { 1/1 a1/l dV
dt
10 at
= -2L1/I (V1/I . v) dV
= -L v· (1/I2v) - 1/I2V . v dV
=0.
The first term in the final integrand is zero by the assumed boundary eonditions;
the seeond term is zero beeause the flow is nondivergent.
The eonservation of 111/1 112 may be used to formulate stability eonditions for
numerical approximations to (3.102) by demanding that the finite-differenee approximation eonserve the diserete analogue of 111/1112. Let u be a eolumn veetor
eontaining the approximate solution at eaeh spatial grid point and A a matrix
eontaining the finite-differenee approximation to v . V1/1 . Then the approximate
solution satisfies a set of linear differential-differenee equations of the form
du
-+Au=O.
dt
(3.104)
This system will eonserve lIull2 if the matrix A is skew-symrnetric, as may be
verified by noting that if A = -AT, then
Assuming that cf> , u, and v are colocated on an unstaggered mesh in a spatially
periodic domain, the nonaveraging operator yields the following differential-differenee approximation to (3.102):
(3.105)
and the averaging operator gives
dcf> + ((u)X oxcf»X + ((v)Y Oycf» Y = O.
dt
(3.106)
Expressing (3.106) in the form (3.104) shows that the matrix A generated by the
averaging sehe me is skew-symmetrie whenever the velocities satisfy
(3.107)
whieh is the diserete analogue of (3.103). On the other hand , the differential-difference approximation (3.105) generated by the nonaveraging operator does not