Conservative Integrators for Piecewise Smooth Systems with Transversal Dynamics

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 Mannshardts 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 integrators is preserved after crossing the discontinuity in the case of codimension one number of conserved quantities. Numerical examples illustrate the preservation of accuracy order.