Characteristic-based treatment of source terms in Euler equations for Roe scheme

The modifications required to apply Roe's Riemann solver to the Euler equations with source terms are developed and demonstrated. The generalized quasi-one-dimensional Row including the effects of friction, secondary mass addition, and energy addition has been considered, The modifications require extensions of the expressions for the strengths of the characteristic waves, The modified expressions for the wave strengths in Roe's Riemann solver are obtained from the theory of characteristics using the compatibility relations that depend on the specific source terms present. It is also shown that appropriate discretization of the source terms is required to obtain the correct solution. It is demonstrated that the compatibility relations in the presence of source terms can be used to modify not only Roe's Riemann solver but also characteristic-based boundary conditions. The validity of the proposed approach is illustrated by comparing the exact analytical solutions and the numerical solutions with and without the proposed corrections for two examples involving steady Bow and one involving unsteady Bow.