  Isogeometric analysis was applied very successfully to many problem classes
like linear elasticity, heat transfer and incompressible flow problems but its
application to compressible flows is very rare. However, its ability to
accurately represent complex geometries used in industrial applications makes
IGA a suitable tool for the analysis of compressible flow problems that require
the accurate resolution of boundary layers. The convection-diffusion solver
presented in this chapter, is an indispensable step on the way to developing a
compressible flow solver for complex viscous industrial flows. It is well known
that the standard Galerkin finite element method and its isogeometric
counterpart suffer from spurious oscillatory behaviour in the presence of
shocks and steep solution gradients. As a remedy, the algebraic flux correction
paradigm is generalized to B-Spline basis functions to suppress the creation of
oscillations and occurrence of non-physical values in the solution. This work
provides early results for scalar conservation laws and lays the foundation for
extending this approach to the compressible Euler equations.
