Computational error bounds for a differential linear variational inequality

The differential linear variational inequality consists of a system of n ordinary differential equations (ODEs) and a parametric linear variational inequality as the constraint. The right-hand side function in the ODEs is not differentiable and cannot be evaluated exactly. Existing numerical methods provide only approximate solutions. In this paper we present a reliable error bound for an approximate solution xh(t) delivered by the time-stepping method, which takes all discretization and roundoff errors into account. In particular, we compute two trajectories xjh(t)±εjh(t) to determine the existence region of the exact solution xj(t),ie.,xjh(t) ≤ xj(t) ≤ xjh(t) + ∈jh(t) for each j ∈ {1,⋯,n}. Moreover, we have ∈jh(t) = O(h). Numerical examples of bridge collapse, earthquake-induced structural pounding and circuit simulation are given to illustrate the efficiency of the error bound.