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 where 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 in two and three dimensions illustrate the preservation of accuracy order across the interface for cubic and logarithmic type conserved quantities. We observed that conservative transition schemes can prevent spurious transitions from occurring, even in the case when there are fewer conserved quantities.