A sign matrix based scheme for non-homogeneous PDE’s with an analysis of the convergence stagnation phenomenon

This work is devoted to the analysis of a finite volume method recently proposed for the numerical computation of a class of non-homogenous systems of partial differential equations of interest in fluid dynamics. The stability analysis of the proposed scheme leads to the introduction of the sign matrix of the flux jacobian. It appears that this formulation is equivalent to the VFRoe scheme introduced in the homogeneous case and has a natural extension here to non-homogeneous systems. Comparative numerical experiments for the Shallow Water and Euler equations with source terms, and a model problem of two-phase flow (Ransom faucet) are presented to validate the scheme. The numerical results present a convergence stagnation phenomenon for certain forms of the source term, notably when it is singular. Convergence stagnation has been also shown in the past for other numerical schemes. This issue is addressed in a specific section where an explanation is given with the help of a linear model equation, and a cure is demonstrated.