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

Conservative Integrators for Piecewise Smooth Systems with Transversal Dynamics