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. We do not perform a rigorous analysis about the convergence or accuracy for the method, the evidence of its effectiveness is presented by means of simulations for physical and synthetical problems.

我们研究一阶偏微分代数方程（PDAE）的系统。对于存在于表面\（\ mathcal {S} \）上的变量，扩展了双曲型偏微分方程理论的经典解，如间断（冲击和接触间断），稀疏度和扩散行波，定义为代数方程组。我们在这里提出一种替代公式，通过将代数条件转换为带有松弛源项（PDRE）的偏微分方程，在数值上和理论上研究PDAE。事实证明，这种松弛系统的解趋向于表面\（\ mathcal {S} \），即长期满足代数方程。我们为PDAE和PDRE系统制定了统一的数值方案。该方案自然是可并行化的，并且收敛速度更快。我们没有对该方法的收敛性或准确性进行严格的分析，其有效性的证明是通过对物理和综合问题的仿真来提供的。