Gautschi-type and implicit-explicit integrators for constrained wave equations

TL;DR

This paper develops and analyzes two second-order integrators, a Crank-Nicolson scheme and a Gautschi-type exponential integrator, for constrained semi-linear wave equations, combining wave and differential-algebraic equation techniques.

Contribution

It introduces novel integrators for constrained wave equations by merging wave integrators with DAE techniques, providing efficient and provably second-order schemes.

Findings

Both integrators are second-order accurate.

The Gautschi-type method involves solving saddle point problems.

Numerical experiments confirm theoretical accuracy and efficiency.

Abstract

This paper deals with the construction and analysis of two integrators for (semi-linear) second-order partial differential-algebraic equations of semi-explicit type. More precisely, we consider an implicit-explicit Crank-Nicolson scheme as well as an exponential integrator of Gautschi type. For this, well-known wave integrators for unconstrained systems are combined with techniques known from the field of differential-algebraic equations. This results in efficient time stepping schemes that are provable of second order. Moreover, we discuss the practical implementation of the Gautschi-type method, which involves the solution of certain saddle point problems. The theoretical results are verified by a numerical experiment for the wave equation with kinetic boundary conditions.