this is what my dad wrote to me:
The flow equations - one continuity and three momentum equations,
also called the Navier-Stokes equations- are non-linear equations.
There are several ways of handling these even in finite difference
and one-dimensional formulation:
1. time-dependent formulation the equations are hyperbolic (in more
than one-dimension, or parabolic for one-dimensional case), to
advance in time (the method is applicable also for steady state
applications till the solutions converge from a given initial
guessed
solution). The method can be implicit or explicit, but one has to
handle non-limear equation in space). The above equations have
to be for compressible case, since the incompressible continuity
equation has no time-dependent term and can not be used to
pressure; one can then use in 1-D case only momentum equation
and Bernoulli's equation to determine pressure and solve the
momentum equation (again non-linear).
2. The above equations can be made linear if, during iteration the
new values = old values + small change in the value. Dropping
the square of (small change in the value) terms, one can get
linearized equations, which can be solved again by implicit or
explicit finite difference methods;
The above is only in a nut-shell. CFD is a too big subject to reply
in one or two words.
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