The class of  Fluctuation Splitting or Residual Distribution schemes has been introduced in the past decade for the solution of steady hyperbolic systems of conservation laws starting from ideas firstly proposed by P.L. Roe and then further developed by Roe and many others.  Although the method has been shown to perform better then more  traditional Finite Volume methods on a wide range of steady homogeneous problems,  its extension to the inhomogeneous case is still an open topic.

As a matter of fact, the method can be easily written in 1D including  consistently the forcing terms, but in the multidimensional case it is not clear  how to include the sources in the first order positive schemes which are at the basis  of the construction of the second order non-linear schemes.  Moreover, it is not yet clear weather upwinding the sources along with the advective  residual would lead to better performances than pointwise or central treatments.

In this work we will focus our attention on the solution of steady scalar  inhomogeneous problems. In particular, an advection-reaction scalar model will be used to give some theoretical basis for the construction of monotone first order discretizations. Concerning the upwinding of the sources, it will be shown that for all the second order linear schemes a consistent discretization leads to
an upwind biased treatment. A monotone first order upwind scheme for inhomogeneous  equations will also be proposed. The numerical experiments show that a considerable reduction  of the error can be achieved by upwinding the source terms. Non linear monotone and almost second  order schemes will be introduced, based on a simple blending of linear schemes.