Parareal for Higher Index Differential Algebraic Equations

TL;DR

This paper adapts the Parareal algorithm for higher index differential algebraic equations by focusing on differential components and using implicit Euler, demonstrating convergence in nonlinear index 2 DAEs.

Contribution

It introduces modifications to Parareal for higher index DAEs, applying it to differential parts and computing consistent initial conditions, with convergence results for specific structures.

Findings

Parareal converges for certain higher index DAEs

Implicit Euler suffices as a time integrator in specific cases

Numerical examples confirm the method's effectiveness

Abstract

This article proposes modifications of the Parareal algorithm for its application to higher index differential algebraic equations (DAEs). It is based on the idea of applying the algorithm to only the differential components of the equation and the computation of corresponding consistent initial conditions later on. For differential algebraic equations with a special structure as e.g. given in flux-charge modified nodal analysis, it is shown that the usage of the implicit Euler method as a time integrator suffices for the Parareal algorithm to converge. Both versions of the Parareal method are applied to numerical examples of nonlinear index 2 differential algebraic equations.