In a semi-implicit method, the pressure is calculated using the conservation equations coupled with the scalar equations. what it the difference between them. how the scalar equations are obtained ?
I don't think you are comparing properly the terms "scalar" and "conservative".
A scalar equation is a single equation for a scalar variable (such as the density) or one component of a vector equation (such as the momentum).
A scalar equation can be also written in conservative form, being this latter property linked to the conservation of the variable in the whole domain.
As example, the density equation is scalar
D rho/Dt = -rho*div v
but can be written also in conservative form
d rho/dt + div (rho*v)=0
since can be integrated over the volume and rewritten as
d/dt Int[V] rho dV = - Int[BV] n.v*rho dS
so that the variation of the total density depends only on the contribution of the mass flux through the boundaries of the volume.
For incompressible flows, the pressure equation is derived by the divergence-free constraint that is a scalar and conservative equation.
These two are not in the same point of view in CFD, you are not comparing apple to apple:
# "Scalar" usually refer to passive scalar transport, it means a passive constituent as opposed to "reactive transport". Like you inject a tracer which does not interact with the surrounding environment and only advects, diffuses and decays (transported) in the media
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# "Conservative" form of a governing equations (as opposed to "Non-conservative" form equation) is a way of forming the equations in which you enforce conserving a property (for example mass) , please look at the screen shoot I attached from Wikipedia. In the conservative form you have "Rho" inside to make sure you conserve mass but in the "non-conservative" form of Shallow Water Equations you drop the "rho"
PS: I have seen in CFD literature sometime people used word "primitive" form of equations instead of "non-conservative"
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