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 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.

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具有横向动力学的分段平滑系统的保守积分器

我们引入了保守积分器，用于具有横向动力学和分段平滑守恒量的分段平滑系统的长期积分。本质上，对于具有分段定义的守恒量的分段动力系统，使得其轨迹横切于其界面，我们将 Mannshardt 的过渡方案和离散乘法器方法相结合，以获得能够保持守恒量达到机器精度和准确度阶数的保守积分器。我们证明，在余维为多个守恒量的情况下，在跨越不连续性后，积分器的精度顺序得以保留。数值例子说明了精度顺序的保持。