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# Conservation equations

The governing equations describing the 1-D non-homentropic gas flow, with the consideration of area change, friction and heat transfer in a pipe, form a non-homogeneous hyperbolic system .

This conservation law system, composed by the continuity, momentum and energy equations, is complemented by the equation of state or the real gas properties . In previous equation, is the desired state vector of the solution, is the flux vector and the source term separating the effect of the area changes from the effect of friction and heat transfer. The 1-D gas flow governing equations were traditionally arranged in the vector form:

# Lax-Wendroff scheme

The Lax-Wendroff scheme is a centred second-order accuracy numerical scheme in which the flow is approximated by Taylor series. OpenWAM applies the two-steps version proposed by Richtmyer and Morton :

# TVD scheme

The TVD flux limiter schemes were presented by Sweby and consist of a first order flux combined with a limited second order flux. Davis and Yee also worked on these techniques approaching homentropic flow. In these flow conditions, the Davis flux limiter technique can be obtained just by adding a viscous term to the second step of Ritchmyer and Morton scheme.

OpenWAM uses an adaptation of the Sweby TVD flux limiter scheme adapted by Gascón . The way to adapt the schemes was to include the source term as part of the flux. The scheme can be written as:

Where the second order accuracy flux can be calculated by means of the expression