Newton iterations in implicit time-stepping scheme for differential linear complementarity systems
The authors consider the following ordinary differential linear complementarity sytem: \[ \begin{gathered} dx/dt= Ax(t)+ By(t)+ f(t),\\ y(t)^T(Nx(t)+ g(t)+ My(t)= 0,\quad y(t)\geq 0,\\ x(0)= x_0,\quad t\in [0,T],\end{gathered} \] where \(A\in\mathbb{R}^{m\times m}\), \(B\in \mathbb{R}^{m\times n}\), \(N\in\mathbb{R}^{n\times m}\), \(M\in\mathbb{R}^{n\times n}\) are given matrices and \(f: \mathbb{R}\to\mathbb{R}^m\), \(g: \mathbb{R}\to \mathbb{R}^n\) are two given Lipschitz continuous functions. The authors propose a time-stepping method for solving the system. The method uses a finite-difference formula to approximate the derivative \(dx(t)/dt\). A Lipschitz continuous solution function \(y(.)\) is found. A simple formula to compute an element in the Clarke generalized Jacobian of the solution function is derived. A sharp and computable Lipschitz constant of \(y(.)\) is prescribed. A superlinear convergence rate of the proposed time-stepping scheme for a class of the given differential linear complementarity system is proved.
differential linear complementarity problem
scientific article; zbMATH DE number 6160896
scientific article