Implicit Runge-Kutta methods for Lipschitz continuous ordinary differential equations

Abstract

Implicit Runge–Kutta (IRK) methods for solving the nonsmooth ordinary differential equation (ODE) involve a system of nonsmooth equations. We show superlinear convergence of the slanting Newton method for solving the system of nonsmooth equations. We prove the slanting differentiability and give a slanting function for the involved function. We develop a new code based on the slanting Newton method and the IRK method for nonsmooth ODEs arising from structural oscillation and pounding. We show that the new code is efficient for solving a nonsmooth ODE model for the collapse of the Tacoma Narrows suspension bridge and simulating 13 different earthquakes.