Conservative Integrators for Piecewise Smooth Systems with Transversal Dynamics

TL;DR

This paper develops conservative numerical integrators for piecewise smooth systems with transversal dynamics, ensuring preservation of conserved quantities with high accuracy during interface crossings.

Contribution

It combines Mannshardt's transition scheme and the Discrete Multiplier Method to create integrators that preserve conserved quantities up to machine precision in piecewise smooth systems.

Findings

Integrators preserve conserved quantities with machine precision.

Accuracy order is maintained after crossing interfaces.

Numerical examples confirm theoretical properties.

Abstract

We introduce conservative integrators for long term integration of piecewise smooth systems with transversal dynamics and piecewise smooth conserved quantities. In essence, for a piecewise dynamical system with piecewise defined conserved quantities such that its trajectories cross transversally to its interface, we combine Mannshardt's transition scheme and the Discrete Multiplier Method to obtain conservative integrators capable of preserving conserved quantities up to machine precision and accuracy order. We prove that the order of accuracy of the conservative integrators is preserved after crossing the interface in the case of codimension one number of conserved quantities. Numerical examples illustrate the preservation of accuracy order and conserved quantities across the interface.