**Keywords:**Partial Differential-Algebraic Equations (PDAEs)\and Hyperbolic system of equations | Riemann problems | parallelizable numerical schemes

We study systems of partial differential-algebraic equations

(PDAEs) of first order. Classical solutions of the theory of Hyperbolic Partial Differential

equation such as discontinuities (shock and contact discontinuities),

rarefactions and diffusive traveling waves are extended for variables

living on a surface $\mathcal{S}$, which is defined as solution of a set of

algebraic equations. We propose here an alternative formulation to study

numerically and theoretically the PDAEs by changing the algebraic conditions into

partial differential equations with relaxation source terms (PDREs).

The solution of such relaxed systems is proved to tend to the surface $\mathcal{S}$, i.e., to satisfy the algebraic equations for long times.

We formulate a unified numerical scheme for systems of PDAEs and PDREs. This scheme is naturally parallelizable and has faster convergence.

Evidence of its effectiveness is presented by means of simulations for physical and synthetic problems.