TVD schemes for the calculation of flow in pipes of variable cross-section

In this paper, first and second order explicit TVD finite difference schemes have been adapted to the calculation of the unsteady one-dimensional flow in ducts of varying cross-sectional area. The basic idea to bear in mind has been the formulation of the schemes in function of the flux instead of the conserved variables, in order to include the term concerning the cross-section variation correctly. The calculation of eigenvalues of Jacobian matrix has been based on the Roeʹs linearization technique, and a modification of the first and third eigenvalues has been done in order to force the satisfaction of the entropy condition. Numerical results are presented to demonstrate the performance of the proposed scheme.