A Gauss--Seidel Type Method for Dynamic Nonlinear Complementarity Problems

The dynamic nonlinear complementarity problem (DNCP) consisting of a nonlinear differential system and a complementarity system has been used to formulate and study many dynamic problems. In a Gauss-Seidel type method for DNCPs, by first guessing a solution of the differential system, we can solve the complementarity system and then with the computed solution we can solve the differential system to update the guess. Upon convergence at the current time point we can move to the next one. The idea can be easily generalized to a multipoint version: instead of doing iterations at each single time point, we can do iterations for a number of time points, say J time points, all at once. Despite its simplicity and easy implementation, convergence of this method is not justified so far. In this paper, we present interesting convergence theorems for this method. We show that the method with a fixed length of time interval converges superlinearly and the convergence rate is robust with respect to the step-size h. Moreover, we show that the method with a fixed number of time points converges with a rate \scrO (h). Since at each iteration the differential system and the complementarity system are solved separately, many existing solvers are directly applicable for each of these two systems. It is notable that we can solve the complementarity system at all the J time points in parallel. Numerical results of the method to solve the 4-diode bridge wave rectifier with random circuit parameters and the projected dynamic systems are given to support our findings.