7.3 Calculation of the Pressure
171
corrected for if necessary) and demand that the divergence at the new time
step n + 1 also be zero. This leads to the Poisson equation for the pressure:
The problem is that the term on the right hand side cannot be computed
until the computation of the velocity field at time n + 1 is completed and
vice versa. As a result, the Poisson equation and the momentum equations
have to be solved simultaneously. That can only be done with some type of
iterative procedure.
Next, even if the pressure were known, Eqs. (7.22) are a large system of
non-linear equations which must be solved for the velocity field. The structure
of this system of equations is essentially the same as the structure of the
matrix of the finite-differenced Laplace equation so solving them is far from
a trivial matter. If one wishes to solve them accurately, the best procedure
is to adopt the converged results from the preceding time step as the initial
guess for the new velocity field and then converge to the solution a t the new
time step using the Newton-Raphson iteration method or a secant method
designed for systems.
An alternative way of dealing with the non-linearity is constructed by
linearizing the equations about the result a t the preceding time step. If we
write:
then the non-linear term in Eqs. (7.22) can be expressed as:
We expect that, at least for small At, Aui
At a u i / a t l SO the last term
in this equation is second order in At and is smaller in magnitude than the
error made in the time discretization. It can therefore be neglected. If we use
a second order method in time, such as the Crank-Nicolson scheme, this term
would be of the same order as the spatial discretization error and we would
still be justified in neglecting it.
We can then write Eq. (7.22) as:
This method takes advantage of the fact that the non-linearity is quadratic
and removes most of the difficulty arising from it. However, we still need to
solve a large system of linear equations with the structure discussed above.
Direct solution of such a system is too expensive to consider so the solution
171
corrected for if necessary) and demand that the divergence at the new time
step n + 1 also be zero. This leads to the Poisson equation for the pressure:
The problem is that the term on the right hand side cannot be computed
until the computation of the velocity field at time n + 1 is completed and
vice versa. As a result, the Poisson equation and the momentum equations
have to be solved simultaneously. That can only be done with some type of
iterative procedure.
Next, even if the pressure were known, Eqs. (7.22) are a large system of
non-linear equations which must be solved for the velocity field. The structure
of this system of equations is essentially the same as the structure of the
matrix of the finite-differenced Laplace equation so solving them is far from
a trivial matter. If one wishes to solve them accurately, the best procedure
is to adopt the converged results from the preceding time step as the initial
guess for the new velocity field and then converge to the solution a t the new
time step using the Newton-Raphson iteration method or a secant method
designed for systems.
An alternative way of dealing with the non-linearity is constructed by
linearizing the equations about the result a t the preceding time step. If we
write:
then the non-linear term in Eqs. (7.22) can be expressed as:
We expect that, at least for small At, Aui
At a u i / a t l SO the last term
in this equation is second order in At and is smaller in magnitude than the
error made in the time discretization. It can therefore be neglected. If we use
a second order method in time, such as the Crank-Nicolson scheme, this term
would be of the same order as the spatial discretization error and we would
still be justified in neglecting it.
We can then write Eq. (7.22) as:
This method takes advantage of the fact that the non-linearity is quadratic
and removes most of the difficulty arising from it. However, we still need to
solve a large system of linear equations with the structure discussed above.
Direct solution of such a system is too expensive to consider so the solution