Time-dependent flow problems are governed by the following set of ordinary differential equations:
 | (29–1) |
which is subject to initial conditions of the type
 | (29–2) |
is the vector of nodal unknowns
such as velocity, pressure, temperature, viscoelastic extra stresses,
and free surface location. The symbol 
denotes
the time-derivative of 
. The matrices 
and 
are the
mass and stiffness matrices, which may depend on the unknown vector 
. The
vector 
corresponds to the volumetric
forcing function and the natural boundary conditions.
The above equations are solved in Ansys Polyflow by means of
a parabolic time-stepping procedure. Instead of trying to satisfy Equation 29–1 at an arbitrary
time 
, Ansys Polyflow will calculate a solution of Equation 29–1 at a discrete
set of times 
, defined by
 | (29–3) |
 | (29–4) |
where the subscript 
refers to the time step.
From Equation 29–1, the time derivative 
can
be obtained. Consider the function 
:
 | (29–5) |
An approximation of 
is
created using the formula
 | (29–6) |
Also, using the first-order discretization of the first derivative
 | (29–7) |
results in
 | (29–8) |
Different values of 
result in different integration methods
with different accuracy and stability attributes.