340
7. Physically InsignificantFast Waves
then the Poisson equation for p'" is
,,2 m
v P
PO,,= 2!i.t v • Y,
and the velocity field is updated using the relation
yn+1 = V_ 2!i.t"ilp'",
Some technique, such as Asselin time filtering (2.50), must also be used to prevent
time-splitting instability in the leapfrog solution to nonlinear problems.
One difference between this approach and the standard projection method is
that by virtue of the leapfrog time difference, the pressure field that ensures the
nondivergence of yn+ I is the actual pressure at time n St . The pressure must,
nevertheless, be updated at the same point in the integration cycle at which ßn+ I
is obtained in the standard projection method, i.e., partway through the calculation
of yn+ I . Thus, the same problems with implicit coupling between the pressure and
yn+ I arise in both the standard and the leapfrog projection methods. If viscosity
is included in the momentum equations, stability considerations require that the
contribution of viscosity to Ftv, p') be evaluated at time level n - I, so that the
viscous terms are treated using forward differencing over a time interval of 2!i.t .
This is not a particularly accurate way to represent the viscous terms and is not
suitable for highly viscous flow, in which the viscous terms are more efficiently
integrated using implicit-time differences.
7.1.3 Solving the Poisson Equationfor Pressure
Suppose that the Boussinesq equations are to be solved in a two-dimensional x-z
domain and that the velocity and pressure variables are staggered as indicated in
Fig. 3.6 . Approximating the diagnostic pressure equation (7.9) using the standard
five-point finite-difference stencil for the two-dimensional Laplacian, one obtains
8 2 p - n+ 1 + 8 2 p - n+ 1 - _1_ (8 U + 8 w)
x
Z
-!i.t x
z ·
(7.13)
This is an implicit algebraic relation for the
I. If pressure is defined at M grid
points in x and N points in z, an M x N system of linear algebraic equations must
be solved in order to determine the pressure. Let the unknown grid-point values
of the press ure be ordered such that
P= ( PU
-n+1 ,PI,2
- n+ 1 ,· · ·,PI.N,P2,1
- n+ 1 -n+1 , ·· · , PM,N ·
-n+1 )
Then the system may be written as the matrix equation
Ap=f,
(7.14)
in which f is an identically ordered vector containing the numerically evaluated
divergence of V. The matrix A is very sparse , with only five nonzero diagonals.
7. Physically InsignificantFast Waves
then the Poisson equation for p'" is
,,2 m
v P
PO,,= 2!i.t v • Y,
and the velocity field is updated using the relation
yn+1 = V_ 2!i.t"ilp'",
Some technique, such as Asselin time filtering (2.50), must also be used to prevent
time-splitting instability in the leapfrog solution to nonlinear problems.
One difference between this approach and the standard projection method is
that by virtue of the leapfrog time difference, the pressure field that ensures the
nondivergence of yn+ I is the actual pressure at time n St . The pressure must,
nevertheless, be updated at the same point in the integration cycle at which ßn+ I
is obtained in the standard projection method, i.e., partway through the calculation
of yn+ I . Thus, the same problems with implicit coupling between the pressure and
yn+ I arise in both the standard and the leapfrog projection methods. If viscosity
is included in the momentum equations, stability considerations require that the
contribution of viscosity to Ftv, p') be evaluated at time level n - I, so that the
viscous terms are treated using forward differencing over a time interval of 2!i.t .
This is not a particularly accurate way to represent the viscous terms and is not
suitable for highly viscous flow, in which the viscous terms are more efficiently
integrated using implicit-time differences.
7.1.3 Solving the Poisson Equationfor Pressure
Suppose that the Boussinesq equations are to be solved in a two-dimensional x-z
domain and that the velocity and pressure variables are staggered as indicated in
Fig. 3.6 . Approximating the diagnostic pressure equation (7.9) using the standard
five-point finite-difference stencil for the two-dimensional Laplacian, one obtains
8 2 p - n+ 1 + 8 2 p - n+ 1 - _1_ (8 U + 8 w)
x
Z
-!i.t x
z ·
(7.13)
This is an implicit algebraic relation for the
I. If pressure is defined at M grid
points in x and N points in z, an M x N system of linear algebraic equations must
be solved in order to determine the pressure. Let the unknown grid-point values
of the press ure be ordered such that
P= ( PU
-n+1 ,PI,2
- n+ 1 ,· · ·,PI.N,P2,1
- n+ 1 -n+1 , ·· · , PM,N ·
-n+1 )
Then the system may be written as the matrix equation
Ap=f,
(7.14)
in which f is an identically ordered vector containing the numerically evaluated
divergence of V. The matrix A is very sparse , with only five nonzero diagonals.