Mathematics and Numerics for Balance Partial Differential-Algebraic Equations (PDAEs)

Wanderson Lambert*, Amaury Alvarez, Ismael Ledoino, Duilio Tadeu, Dan Marchesin, Johannes Bruining

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

9 Citations (Scopus)


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 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 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.

Original languageEnglish
Article number29
Number of pages56
JournalJournal of Scientific Computing
Issue number2
Publication statusPublished - 2020


  • Hyperbolic system of equations
  • Parallelizable numerical schemes
  • Partial differential-algebraic equations (PDAEs)
  • Riemann problems


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